D-brane Masses at Special Fibres of Hypergeometric Families of Calabi–Yau Threefolds, Modular Forms, and Periods

Abstract

We consider the fourteen families W of Calabi–Yau threefolds with one complex structure parameter and Picard–Fuchs equation of hypergeometric type, like the mirror of the quintic in $\mathbb {P}^4$. Mirror symmetry identifies the masses of even-dimensional D-branes of the mirror Calabi–Yau M with four periods of the holomorphic (3, 0)-form over a symplectic basis of $H_3(W,\mathbb {Z})$. It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under $\Gamma _0(25)$, whose Hecke eigenvalues are determined by the Hasse–Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse–Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga–Sato variety. We also comment on simpler applications to local Calabi–Yau threefolds.

Appendices

Appendix: Modular Forms and Arithmetic Algebraic Geometry

In the first two parts of this appendix we review the general theory of modular forms and their associated period polynomials, which leads to the definition of periods and quasiperiods of modular forms. In the second part we review the cohomological structure of smooth projective varieties, which for example gives rise to periods and zeta functions. We sketch how the different cohomology groups define motives and that also to certain modular forms one can attach motives.

1.1 Cusp forms and periods

In this section we define the periods associated with modular forms for discrete and cofinite subgroups $\Gamma$ of $\textrm{SL}(2,\mathbb {R})$. For us the relevant examples are the level N subgroups $\Gamma _0(N)\subseteq \textrm{SL}(2,\mathbb {Z})$. We start the section by reviewing a few basic facts about these groups and the properties of modular forms. Then we describe how one can associate period polynomials with modular forms and construct these explicitly for the group $\Gamma _0^*(25)$ and weight 4.

1.1.1 Review of holomorphic modular forms

In this section we review some elementary facts about holomorphic modular forms. For further details, see e.g. [83] or [84].

The group ${\textrm{SL}}(2,\mathbb {R})$ of real $2\times 2$ matrices of determinant 1 acts as usual on the complex upper half plane $\mathfrak {H}= \{ \tau \in \mathbb {C}| {\textrm{Im}}\tau > 0 \}$ by $\tau \mapsto g\tau = \frac{a\tau +b}{c\tau +d}$ for $g = \left( \begin{array}{c} a\, b\\ c\, d \end{array}\right) \in {\textrm{SL}}(2,\mathbb {R})$ and this action also extends to $\mathfrak {H}\cup \mathbb {P}^1(\mathbb {R})$. Elements in ${\textrm{SL}}(2,\mathbb {R})$ which have exactly one fixed point in $\mathbb {P}^1(\mathbb {R})$ are called parabolic elements and every parabolic element is conjugate to $\pm T$, where $T = \left( \begin{array}{c} 1\, 1\\ 0\, 1 \end{array}\right)$. Now let $\Gamma$ be a discrete subgroup of ${\textrm{SL}}(2,\mathbb {R})$ that is cofinite, i.e. $\Gamma \backslash \mathfrak {H}$ has finite hyperbolic area. The fixed points in $\mathbb {P}^1(\mathbb {R})$ with respect to parabolic elements of $\Gamma$ are called the cusps of $\Gamma$ and we denote the union of $\mathfrak {H}$ and the set of cusps of $\Gamma$ by $\overline{\mathfrak {H}}$. The action of $\Gamma$ can be restricted to $\overline{\mathfrak {H}}$ and two cusps are said to be equivalent if they are in the same $\Gamma$ orbit. There are only finitely many equivalence classes of cusps.

For any function $f:\mathfrak {H}\rightarrow \mathbb {C}$, integer $k\in \mathbb {Z}$, and $g=\left( \begin{array}{c} a\, b\\ c\, d \end{array}\right) \in {\textrm{SL}}(2,\mathbb {R})$ one writes

$$\begin{aligned} (f|_k g)(\tau )=(c\tau + d)^{-k} f(g\tau ) \end{aligned}$$

(A.1)

and calls $|_k$ the weight k slash operator. For any $k\in \mathbb {Z}$ we define the vector space $M_k(\Gamma )$ of (holomorphic) modular forms by

$$\begin{aligned} M_k(\Gamma ) = \{ f : \mathfrak {H}\rightarrow \mathbb {C}\mid f|_k\gamma = f\; \forall \,\gamma \in \Gamma , f \text { holomorphic on }\overline{\mathfrak {H}} \} , \end{aligned}$$

(A.2)

where f is said to be holomorphic (vanish) at a cusp fixed by $\pm g T g^{-1} \in \Gamma$ if $(f|_kg)(x+iy)$ is bounded (vanishes) for $y \rightarrow \infty$. A modular form $f \in M_k(\Gamma )$ is a cusp form if it vanishes at all cusps. We denote the subspace of cusp forms by $S_k(\Gamma ) \subseteq M_k(\Gamma )$. The spaces $M_k(\Gamma )$ and hence $S_k(\Gamma )$ are finite–dimensional and there are standard formulas for $\dim M_k(\Gamma )$ and $\dim S_k(\Gamma )$.

Modular forms have Fourier expansions around each cusp, i.e. for a cusp fixed by $\pm g T g^{-1} \in \Gamma$ one finds that $(f|_kg)(\tau +1) = (\pm 1)^k(f|_kg)(\tau )$ and hence there is an expansion

$$\begin{aligned} (f|_kg)(\tau ) = \sum _m a_{g,m} \, q^m \qquad \text {with} \qquad q = e^{2\pi i \tau } , \end{aligned}$$

(A.3)

where, depending on $(\pm 1)^k$, the sum runs over positive integers or positive half integers. If f is a cusp form we further have $a_{g,0} = 0$. If $T \in \Gamma$ we abbreviate $a_{1,m}$ by $a_m$ and then have

$$\begin{aligned} f(\tau ) = \sum _{m=0}^\infty a_m \, q^m . \end{aligned}$$

(A.4)

1.1.2 Hecke operators and Atkin–Lehner involutions

From now on we take for $\Gamma$ the level N subgroup

$$\begin{aligned} \Gamma _0(N)=\left. \left\{ \left( \begin{array}{ll} a &{} b\\ c &{} d\end{array}\right) \in {\textrm{SL}}(2,\mathbb {Z}) \;\right| \; c\equiv 0 {\textrm{mod}}N\;\right\} \qquad (N\in \mathbb {N} ) \end{aligned}$$

(A.5)

and for each $n\in \mathbb {N}$ with $(n,N)=1$ define the Hecke operator $T_n$, acting on $M_k(\Gamma _0(N))$, as follows. Let

$$\begin{aligned} \mathcal {M}_{n,N}=\left. \left\{ g=\left( \begin{array}{ll} a &{} b\\ c &{} d\end{array}\right) \in \textrm{M}_2(\mathbb {Z}) \;\right| \; \det (g) = n, c \equiv 0 {\textrm{mod}}N \right\} , \end{aligned}$$

(A.6)

where $\textrm{M}_2(\mathbb {Z})$ denotes the set of integral $2\times 2$ matrices. Note that this set is stabilized under left and right multiplication by any $\gamma \in \Gamma _0(N)$. For $f \in M_k(\Gamma _0(N))$ we then define

$$\begin{aligned} f|_kT_n =n^{k-1}\sum _{M\in \Gamma _0(N)\backslash \mathcal {M}_{n,N}} f |_k M\ , \end{aligned}$$

(A.7)

where the weight k slash operator on the right is defined as in (A.1) even though the matrices M do not have determinant 1. The sum is over any set of representatives for the left action of $\Gamma _0(N)$ on $\mathcal {M}_{n,N}$, a convenient choice being

$$\begin{aligned} \mathcal {M}_{n}^{[\infty ]}=\left. \left\{ \left( \begin{array}{ll} a &{} b\\ 0 &{} d\end{array}\right) \in \textrm{M}_2(\mathbb {Z}) \;\right| \; a d =n, \ 0\le b < d \right\} . \end{aligned}$$

(A.8)

Note that the cardinality of this set equals $\sigma _1(n)$, the sum of divisors of n. In particular, the sum in (A.7) is finite and does not depend on the choice of representatives since f is modular. It is easy to see that $f|_kT_n$ is again modular since the set $\Gamma _0(N)\backslash \mathcal {M}_{n,N}$ is invariant under right multiplication by any $\gamma \in \Gamma _0(N)$. We further see that $T_n$ maps cusp forms to cusp forms. Since $T \in \Gamma _0(N)$ we have the Fourier expansion (A.4) and if one chooses the representatives as in (A.8) one gets a formula for the action of $T_n$ on the Fourier expansion of f. For cusp forms this gives

$$\begin{aligned} (f|_kT_n)(\tau ) =\sum _{m=1}^\infty \sum _{r|(m,n)\atop r>0} r^{k-1}\, a_{mn/r^2}\, q^m. \end{aligned}$$

(A.9)

Using the fact that the $T_n$ for different n commute which each other, and that they are self–adjoint for a certain inner product (the Petersson inner product) on $S_k(\Gamma _0(N))$, one can choose a common basis of eigenforms f of $S_k(\Gamma _0(N))$ such that

$$\begin{aligned} f|_k T_n=\lambda _n f\qquad \forall \;n \in \mathbb {N}, \; (n,N) = 1 . \end{aligned}$$

(A.10)

From (A.9) one then gets $a_n = \lambda _na_1$ for $(n,N)=1$. In particular, for $N=1$ any eigenform is (up to a multiplicative constant) uniquely determined by its Hecke eigenvalues. For $N>1$ this is not true in general but for so called newforms $f\in S_k(\Gamma _0(N))$, which are eigenforms under all Hecke operators that are normalized by $a_1=1$ and that are orthogonal to all $g(m \, \tau )$ for integers m and modular forms $g \in S_k(\Gamma _0(N'))$ with $m \, N' \, | \, N$, this is again true. We denote the algebra generated by the Hecke operators by $\mathbb {T}$.

There is a further set of operators on $M_k(\Gamma _0(N))$ that are relevant for us. For any exact divisor Q of N, i.e. Q|N and $(Q,N/Q) =1$, any element in the set

$$\begin{aligned} \mathcal {W}_Q = \frac{1}{\sqrt{Q}} \begin{pmatrix} Q\mathbb {Z}&{} \mathbb {Z}\\ N\mathbb {Z}&{} Q\mathbb {Z}\\ \end{pmatrix} \cap {\textrm{SL}}(2,\mathbb {R}) \end{aligned}$$

(A.11)

normalizes $\Gamma _0(N)$ and the product of any two elements of $\mathcal {W}_Q$ is in $\Gamma _0(N)$. Hence, any $W_Q \in \mathcal {W}_Q$ induces an involution on $\Gamma _0(N)\backslash \overline{\mathfrak {H}}$ via the action of $W_Q$ on $\overline{\mathfrak {H}}$. These involutions do not depend on the choice of $W_Q \in \mathcal {W}_Q$ and are called the Atkin–Lehner involutions. They generate a group isomorphic to $(\mathbb {Z}/2\mathbb {Z})^{\ell }$, where $\ell$ is the number of prime factors of N. The subgroup of ${\textrm{SL}}(2,\mathbb {R})$ obtained by adjoining all Atkin–Lehner involutions to $\Gamma _0(N)$ is denoted by $\Gamma _0^*(N)$, i.e.

$$\begin{aligned} \Gamma _0^*(N) = \bigcup _{\begin{array}{c} Q\mid N\\ (Q,N/Q)=1 \end{array}} W_Q\Gamma _0(N) . \end{aligned}$$

(A.12)

It normalizes $\Gamma _0(N)$ in ${\textrm{SL}}(2,\mathbb {R})$ and permutes the cusps of $\Gamma _0(N)$. Each Atkin–Lehner involution on $\Gamma _0(N) \backslash \overline{\mathfrak {H}}$ induces an involution (also called Atkin–Lehner involution) on $M_k(\Gamma _0(N))$ by $f \mapsto f|_kW_Q$, which is again independent of the choice of $W_Q$. These involutions commute with each other as well as with the operators of $\mathbb {T}$ and define an eigenspace decomposition $M_k(\Gamma _0(N)) = \bigoplus _{\epsilon } M^{\epsilon }_k(\Gamma _0(N))$, where the sum ranges over the characters of $(\mathbb {Z}/2\mathbb {Z})^\ell$. The fact that the Atkin–Lehner involutions commute with $\mathbb {T}$ implies that every newform automatically belongs to one of these eigenspaces.

1.1.3 Eichler integrals and period polynomials

We consider the normalized derivative $D = \frac{1}{2\pi i} \frac{\text {d}^{}{}}{\text {d}^{}{\tau }}$, where the factor $\frac{1}{2\pi i}$ is introduced so that D sends periodic functions with rational Fourier coefficients to periodic functions with rational Fourier coefficients. The operator D does not preserve modularity. Instead, we have the following elementary but not obvious proposition.

Proposition 1

(Bol’s identity [85]). Let $k \in \mathbb {N}$ be an integer, $k \ge 2$. Then for any meromorphic function $f: \mathfrak {H}\rightarrow \mathbb {P}^1(\mathbb {C})$ we have

$$\begin{aligned} D^{k-1}(f|_{2-k} g)=(D^{k-1} f)|_k g\qquad ( \forall \; g \in {\textrm{SL}}(2,\mathbb {R})) . \end{aligned}$$

(A.13)

If f is modular of weight k on some group $\Gamma$, then any holomorphic function $\widetilde{f}: \mathfrak {H}\rightarrow \mathbb {C}$ with the property that $D^{k-1} {\widetilde{f}} = f$ is called an Eichler integral of f. The Eichler integral exists, but is well–defined only up to a degree $k-2$ polynomial $p \in V_{k-2}(\mathbb {C})$, where $V_{k-2}(K)=\langle 1,\ldots ,\tau ^{k-2}\rangle _{K}$ for any field K. For instance, we can take ${\widetilde{f}}$ to be $\widetilde{f}_{\tau _0}$, where

$$\begin{aligned} \widetilde{f}_{\tau _0}(\tau ) = \frac{(2\pi i)^{k-1}}{(k-2)!} \int _{\tau _0}^\tau (\tau -z)^{k-2}\, f(z) \, \hbox {d}z \end{aligned}$$

(A.14)

for any $\tau _0 \in \mathfrak {h}$, or even $\tau _0 \in {\overline{\mathfrak {h}}}$ if f is a cusp form. In particular, if $T \in \Gamma$, then we have

$$\begin{aligned} \widetilde{f}_\infty (\tau ) = \sum _{m=1}^\infty \frac{a_m}{m^{k-1}}\, q^m \quad \text {if} \quad f (\tau ) = \sum _{m=1}^\infty a_m\, q^m \; \in S_k(\Gamma ). \end{aligned}$$

(A.15)

For later purposes we observe that ${\widetilde{f}}_\infty$ is related to $\widetilde{f}_{\tau _0}$ for any $\tau _0 \in \mathfrak {h}$ by

$$\begin{aligned} \widetilde{f}_\infty (\tau )-\widetilde{f}_{\tau _0}(\tau ) = \frac{(2\pi i)^{k-1}}{(k-1)!} \int _{\tau _0}^{\tau _0-1}B_{k-1}(\tau -z)\, f(z) \, \hbox {d}z , \end{aligned}$$

(A.16)

where $B_n$ is the nth Bernoulli polynomial. Indeed, from $B_n(x+1)=B_n(x)+nx^{n-1}$ and $f(z-1)=f(z)$ we find that this equation does not depend on $\tau _0$ and since it is true for $\tau _0 = \infty$ it is true for all $\tau _0$.

For a fixed choice of Eichler integral $\tilde{f}$ it follows from Bol’s identity (A.13) that

$$\begin{aligned} r_{f}(\gamma )\, := \, {\widetilde{f}}|_{2-k} (\gamma -1)(\tau ) \, \in \, V_{k-2}(\mathbb {C}) \qquad \forall \; \gamma \in \Gamma \end{aligned}$$

(A.17)

i.e. $r_f(\gamma )$ is a polynomial of degree $k-2$, which is called a period polynomial of f for $\gamma \in \Gamma$. Here we extended the action of the slash operator to the group algebra $\mathbb {C}[{\textrm{SL}}(2,\mathbb {R})]$ in the obvious way (viz., $f|_k\sum g_i =\sum f|_kg_i$, where we write $\sum g_i$ instead of the more correct $\sum [g_i]$). The period polynomials measure the failure of modularity of the Eichler integral. An immediate consequence of the definition is that the period polynomials satisfy the cocycle condition

$$\begin{aligned} r_f(\gamma \gamma ') = r_f(\gamma )|_{2-k}\gamma ' + r_f(\gamma ') , \end{aligned}$$

(A.18)

where we define an action of ${\textrm{SL}}(2,\mathbb {Z})$ on $V_{k-2}(\mathbb {C})$ by extending the slash operator (A.1) to complex polynomials $p\in V_{k-2}(\mathbb {C})$ in the obvious way.

Since the Eichler integral ${\widetilde{f}}$ is unique only up to addition of polynomials $p \in V_{k-2}(\mathbb {C})$ it follows that $r_f$ is unique only up to addition of maps of the form $\gamma \mapsto p|_{2-k}(\gamma - 1)$ for polynomials $p \in V_{k-2}(\mathbb {C})$. The dependence on p is described in terms of group cohomology. Let K be any field so that $\Gamma \subset \text {SL}(2,K)$. We define the group of cocycles

$$\begin{aligned} Z^1(\Gamma ,V_{k-2}(K)) = \{ r: \Gamma \rightarrow V_{k-2}(K) \mid r(\gamma \gamma ') = r(\gamma )|_{2-k}\gamma ' + r(\gamma ') \;\forall \; \gamma ,\gamma ' \in \Gamma \} \end{aligned}$$

(A.19)

and the group of coboundaries by

$$\begin{aligned} B^1(\Gamma ,V_{k-2}(K)) = \{ \Gamma \ni \gamma \mapsto p|_{2-k}(\gamma -1) \mid p \in V_{k-2}(K)\} . \end{aligned}$$

(A.20)

Then, the (first) group cohomology group is defined as the quotient

$$\begin{aligned} H^1(\Gamma , V_{k-2}(K)) = \frac{Z^1(\Gamma , V_{k-2}(K))}{B^1(\Gamma ,V_{k-2}(K))} . \end{aligned}$$

(A.21)

It follows from the definition (A.17) that the freedom in the choice of the Eichler integral ${\widetilde{f}}$ results in a coboundary. Therefore we can associate to f a unique cohomology class $[r_f] \in H^1(\Gamma , V_{k-2}(\mathbb {C}))$. Furthermore, we define the group of parabolic cocycles

$$\begin{aligned}{} & {} Z^1_{\textrm{par}}(\Gamma , V_{k-2}(K))\nonumber \\{} & {} \quad =\{r \in Z^1(\Gamma , V_{k-2}(K)) \mid r(\gamma )\in V_{k-2}(K) |_{2-k}(\gamma -1) \ \forall \; \text {parabolic} \; \gamma \in \Gamma \} .\nonumber \\ \end{aligned}$$

(A.22)

Trivially, one has ${B^1 \subseteq Z^1_{\textrm{par}}\subseteq Z^1}$. Hence, one can define the parabolic cohomology group

$$\begin{aligned} H^1_{\textrm{par}}(\Gamma , V_{k-2}(K)) = \frac{Z^1_{\textrm{par}}(\Gamma , V_{k-2}(K))}{B^1(\Gamma ,V_{k-2}(K))} \, \subseteq \, H^1(\Gamma , V_{k-2}(K)) , \end{aligned}$$

(A.23)

where the codimension of the embedding is in general less or equal then the number of non-equivalent cusps times the dimension of $V_{k-2}(K)$. We have the following proposition.

Proposition 2

For any $f \in S_k(\Gamma )$ one has $r_f \in Z^1_{\textrm{par}}(\Gamma , V_{k-2}(\mathbb {C}) )$.

Proof

Recall that $r_f$ is defined by (A.17) for some fixed Eichler integral ${\widetilde{f}}$ of f. We have to show that $r_f(\gamma )$ belongs to $V_{k-2}(\mathbb {C})|(\gamma -1)$ for any parabolic $\gamma \in \Gamma$. We can write $\gamma = \pm gTg^{-1} \in \Gamma$ for some $g \in {\textrm{SL}}(2,\mathbb {R})$. Then we have a Fourier expansion

$$\begin{aligned} (f|_kg)(\tau ) = \sum _m a_{g,m} \, q^m , \end{aligned}$$

(A.24)

where $a_{g,0}$ vanishes since f is a cusp form. The function

$$\begin{aligned} \biggl (\sum _m \frac{a_{g,m}}{m^{k-1}}q^m \biggr )\Big |_{2-k}g^{-1} \end{aligned}$$

(A.25)

is then annihilated by $\gamma -1$, and using Bol’s identity we find that it is an Eichler integral of f and hence differs from ${\widetilde{f}}$ by an element of $V_{k-2}(\mathbb {C})$. This implies the claim. $\square$

The importance of the parabolic cohomology group stems from a theorem due to Eichler. We define the space of $\overline{S_k(\Gamma )}$ of antiholomorphic cusp forms as the space of all functions $\overline{f}$ for $f \in S_k(\Gamma )$, where we define $\overline{f}(\tau ) = \overline{f(\tau )}$.

Theorem 1

(Eichler–Shimura isomorphism). The map $f \mapsto [r_f]$ and its complex conjugate $\overline{f} \mapsto [r_{\overline{f}}]:= [\overline{r_{f}}]$ (obtained by complex conjugating the coefficients) induce an isomorphism

$$\begin{aligned} H^1_{\textrm{par}}(\Gamma , V_{k-2}(\mathbb {C}) ) \, \cong \, S_k(\Gamma ) \oplus \overline{S_k(\Gamma )} . \end{aligned}$$

(A.26)

Proof

For even k a first result of this type was given by Eichler in [86], who in particular showed that the dimensions of both sides agree. For the complete proof for even and odd k we refer to Shimura [87]. $\square$

We now assume that $\varepsilon = \left( \begin{array}{c} -1\, 0\\ 0\,1 \end{array}\right)$ normalizes $\Gamma$. We then get an involution $r \mapsto r|_{2-k}\varepsilon$ on $Z^1(\Gamma ,V_{k-2}(K))$, where we define the action of any normalizer $W \in \text {GL}(2,K)$ of $\Gamma$ on elements in $Z^1(\Gamma ,V_{k-2}(K))$ by

$$\begin{aligned} (r|_{2-k}W)(\gamma ) = r(W\gamma W^{-1})|_{2-k}W. \end{aligned}$$

(A.27)

Here we generalize that the slash operator acts on polynomials as defined in (A.1) even when $\det W <0$. The eigenvalues of the involution are $\pm 1$ and we get an induced decomposition

$$\begin{aligned} H^1_{\textrm{par}}(\Gamma , V_{k-2}(K) ) = H^1_{\textrm{par}}(\Gamma , V_{k-2}(K) )^+ \oplus H^1_{\textrm{par}}(\Gamma , V_{k-2}(K) )^-. \end{aligned}$$

(A.28)

It is straightforward to check that, with respect to the Eichler-Shimura isomorphism, the involution $\varepsilon$ on $H^1_{\textrm{par}}(\Gamma , V_{k-2}(\mathbb {C}) )$ corresponds to the involution on $S_k(\Gamma ) \oplus \overline{S_k(\Gamma )}$ induced by ${f \mapsto (-1)^{k-1}f^*}$, where $f^*(\tau ) = f(-\overline{\tau })$. In particular, the restriction of period polynomials to $H^1_{\textrm{par}}(\Gamma , V_{k-2}(K) )^\pm$ gives the isomorphisms

$$\begin{aligned} S_k(\Gamma ) \cong H^1_{\textrm{par}}(\Gamma , V_{k-2}(\mathbb {C}) )^{\pm } . \end{aligned}$$

(A.29)

We now fix $\Gamma =\Gamma _0(N)$. Since $S_k(\Gamma _0(N))$ admits an action by the Hecke algebra $\mathbb {T}$, the Eichler-Shimura isomorphism induces an action of $\mathbb {T}$ on $H^1_{\textrm{par}}(\Gamma _0(N), V_{k-2}(\mathbb {C}) )$. This action can be described as follows. For a map $r: \Gamma _0(N) \rightarrow V_{k-2}(K)$ and for $n \in \mathbb {N}$ with $(n,N)=1$ we define a map $r|_{2-k}T_n: \Gamma _0(N) \rightarrow V_{k-2}(K)$ by

$$\begin{aligned} (r|_{2-k}T_n)(\gamma ) = \sum _{i=1}^{\sigma _1(n)} r(\gamma _i)|_{2-k} M_{\pi _\gamma (i)} , \end{aligned}$$

(A.30)

where $M_i$, $i=1,\dots ,\sigma _1(n)$ are chosen representatives of $\Gamma _0(N) \backslash \mathcal {M}_{n,N}$ and the $\gamma _i \in \Gamma _0(N)$ are determined by the identity

$$\begin{aligned} M_i \gamma = \gamma _i M_{\pi _\gamma (i)} . \end{aligned}$$

(A.31)

Here, $\pi _\gamma (i)$ denotes a permutation of the indices $i=1,\dots ,\sigma _1(n)$, whose dependence on $\gamma$ is uniquely determined by (A.31). Using the cocycle property it is straightforward to show that this map can be restricted to $Z^1$ and $B^1$. Further, the map depends on the choice of representatives of $\Gamma _0(N) \backslash \mathcal {M}_{n,N}$, but we have the following propositions.

Proposition 3

For any $r \in Z^1(\Gamma _0(N),V_{k-2}(K))$ the cohomology class $[r|_{2-k}T_n]$ does not depend on the chosen representatives of $\Gamma _0(N) \backslash \mathcal {M}_{n,N}$.

Proof

Let $r|_{2-k}T_n'$ be defined with respect to a second choice $M_i'$, $i=1,...,\sigma _1(n)$ of representatives of $\Gamma _0(N) \backslash \mathcal {M}_{n,N}$. We order these so that $M_i'=\gamma '_i M_i$ for uniquely determined $\gamma '_i \in \Gamma _0(N)$. By using the cocycle property one finds that for all $\gamma \in \Gamma _0(N)$

$$\begin{aligned} (r|_{2-k}T_n' - r|_{2-k}T_n)(\gamma ) = \Big ( \sum _{i=1}^{\sigma _1(n)} r(\gamma '_i)|_{2-k}M_i \Big )|_{2-k} (\gamma - 1) \end{aligned}$$

(A.32)

and thus $[r|_{2-k}T_n']=[r|_{2-k}T_n]$. $\square$

Proposition 4

For $f\in S_k(\Gamma _0(N))$ we have

$$\begin{aligned} r_{f|_kT_n} = r_f|_{2-k}T_n , \end{aligned}$$

(A.33)

where the same set of representatives of $\Gamma _0(N) \backslash \mathcal {M}_{n,N}$ has been chosen on both sides and the Eichler integral on the left side has been chosen as $\widetilde{f|_kT_n} = n^{k-1}\widetilde{f}|_{2-k}T_n$.

Proof

Using Bol’s identity (A.13) we find that

$$\begin{aligned} D^{k-1}(n^{k-1}\widetilde{f}|_{2-k}T_n) = (D^{k-1}\widetilde{f})|_kT_n = f|_kT_n \end{aligned}$$

(A.34)

and thus our choice of Eichler integral is indeed valid. We then get

$$\begin{aligned} r_{f|_kT_n}(\gamma )&= \left. \widetilde{f|_kT_n}\right| _{2-k}(\gamma -1) \nonumber \\&= n^{k-1}\left. \widetilde{f}|_{2-k}T_n\right| _{2-k}(\gamma -1)\nonumber \\&=\sum _{i=1}^{\sigma _1(n)} \widetilde{f}|_{2-k}(M_i\gamma -M_i) \nonumber \\&= \sum _{i=1}^{\sigma _1(n)} \widetilde{f}|_{2-k}(\gamma _iM_{\sigma _\gamma (i)}-M_i) \nonumber \\&= \sum _{i=1}^{\sigma _1(n)} r_f(\gamma _i)|_{2-k}M_{\sigma _\gamma (i)} \end{aligned}$$

(A.35)

$\square$

We conclude that the action of $\mathbb {T}$ defined by (A.30) induces a well defined action of Hecke operators on $H^1(\Gamma _0(N),V_{k-2}(K))$ which does not depend on the chosen representatives of $\Gamma _0(N) \backslash \mathcal {M}_{n,N}$ and is compatible with the isomorphism (A.26) for $K=\mathbb {C}$. Completely analogously we can define the action of Atkin–Lehner operators $W_Q$ on $Z^1(\Gamma _0(N),V_{k-2}(K))$ (for suitable K) by $r \mapsto r|_{2-k}W_Q$. This gives a well-defined action on $H^1(\Gamma _0(N),V_{k-2}(K))$ which does not depend on the chosen element of $\mathcal {W}_Q$ and is compatible with the isomorphism (A.26) for $K=\mathbb {C}$.

We conclude this introduction to period polynomials with an important proposition about the period polynomials associated with newforms.

Proposition 5

Let $f \in S_k(\Gamma _0(N))$ be a newform and let $\mathbb {Q}(f)$ be the number field generated by its Hecke eigenvalues. Then the Eichler integral can be chosen such that

$$\begin{aligned} r_f \, \in \, \omega _f^+Z^1_{\textrm{par}}(\Gamma _0(N), V_{k-2}(\mathbb {Q}(f)))^+\oplus \omega _f^-Z^1_{\textrm{par}}(\Gamma _0(N), V_{k-2}(\mathbb {Q}(f)))^- \end{aligned}$$

(A.36)

for some $\omega _f^\pm \in \mathbb {C}$. If $\mathbb {Q}(f)$ is totally real one has $\omega _f^+ \in \mathbb {R}$ and $\omega _f^- \in i\mathbb {R}$.

Proof

First note that $H^1_{\textrm{par}}(\Gamma _0(N), V_{k-2}(\mathbb {C})) \cong H^1_{\textrm{par}}(\Gamma _0(N), V_{k-2}(\mathbb {Q})) \otimes _{\mathbb {Q}} \mathbb {C}$ and that we have a well-defined action of the Hecke algebra $\mathbb {T}$ and of the involution $\varepsilon$ on $H^1_{\textrm{par}}(\Gamma _0(N), V_{k-2}(\mathbb {Q}))$. Since f is uniquely determined by its Hecke eigenvalues which lie in $\mathbb {Q}(f)$ we can define two 1-dimensional eigenspaces $V^{\pm } \subseteq H^1_{\textrm{par}}(\Gamma _0(N), V_{k-2}(K))^{\pm }$ with the same eigenvalues as $f\pm (-1)^{k-1}f^*$. Then the first statement directly follows. If $\mathbb {Q}(f)$ is totally real we have $f^*=\overline{f}$ and then the second statement also follows. $\square$

We call the numbers $\omega _f^{\pm }$, which are unique only up to multiplication by $\mathbb {Q}(f)$, the periods of f. Proposition 5 was first proved (for $\Gamma = {\textrm{SL}}(2,\mathbb {Z})$) by Manin [88] in a stronger form, namely that the period polynomials $r_f$ defined by choosing ${\widetilde{f}} = {\widetilde{f}}_\infty$ as in (A.15) satisfies (A.36), and we will use this in the sequel.

1.1.4 Computation of $H^1_{\text {par}}(\Gamma _0^*(25),V_2(\mathbb {Q}))$

In the following we want to compute a basis for $H^1_{\textrm{par}}(\Gamma _0^*(25),V_2(\mathbb {Q}))$ and simultaneously diagonalize the action of the involution $\varepsilon$ and the Hecke algebra $\mathbb {T}$. We start by explaining how one can obtain a set of generators of $\Gamma _0^*(25)$ and their relations.

To obtain generators of discrete cofinite subgroups $\Gamma \subseteq \text {SL}(2,\mathbb {R})$ we construct a fundamental domain $\mathcal {F}$ as $\mathcal {F}_{\tau {}_0}$ plus parts of its boundary, where

(A.37)

Here $\tau _0\in \overline{\mathfrak {H}}$ is arbitrary point, d is the hyperbolic distance function and $\Gamma _{\tau _0}$ is the stabilizer of $\tau _0$. If $\overline{\mathfrak {H}}$ contains $\infty$, choosing $\tau _0=\infty$ and the Ford circles $|c \tau + d|= 1$ as boundaries is particularly convenient. Then (A.37) evaluates to

(A.38)

For subgroups of ${\textrm{SL}}(2,\mathbb {R})$ containing $T = \left( \begin{array}{c} 1\, 1\\ 0\, 1 \end{array}\right)$ we define $\mathfrak {H}_s$ to be the strip of width 1 in the upper half plane $\mathfrak {H}_s=\left\{ \tau \in \mathfrak {H}\mid -\frac{1}{2}< {\textrm{Re}}\tau < \frac{1}{2}\right\}$. For instance, for $\Gamma = \Gamma _0(N)$ one can then express (A.38) as

(A.39)

while for $\Gamma _0^*(N)$ with N a prime power we instead take the product over all elements that are in $W_N \Gamma _0(N)$ (or $\Gamma _0(N) W_N$), i.e. over elements of the form $\left( \begin{array}{c} {\hat{a}} \sqrt{N} \;\; {\hat{b}}/\sqrt{N}\\ {\hat{c}} \sqrt{N} \;\;\; {\hat{d}} \sqrt{N} \end{array}\right)$ with ${\hat{a}},{\hat{b}},{\hat{c}},{\hat{d}}\in \mathbb {Z}$ and $N {\hat{a}} {\hat{d}} -{\hat{b}}{\hat{c}}=1$, hence the divisibility condition becomes $({\hat{c}}, N {\hat{d}})=1$. The union over c leads to Ford circles with rapidly decreasing radii, which can be shown to not bound the fundamental domain further for c sufficiently large. Topologically $\mathcal {F}$ is a polygon bounded by segments of the Ford circles as edges. If a Ford circle has a fixed point of order 2 we regard it as two edges split by the fixed point. In this way the polygon has an even number of edges which are identified in pairs. As generators of $\Gamma$ one can choose the elements identifying the edges. The relations between these generators are obtained from considering the finite orbits of the vertices of $\mathcal {F}$ under the action of $\Gamma$. If the vertices in one orbit are cusps, one gets no relation, and if they are elliptic points of order n, one gets a product of elements which is of order n (in $\Gamma / \{\pm 1\}$). As an example, consider the standard fundamental domain of ${\textrm{SL}}(2,\mathbb {Z})$ with the vertices $P_0 =\infty , P_1=e^{2\pi i/3}, P_2=i, P_3=e^{\pi i/3}$. The edges $P_0P_1$ and $P_3P_0$ are identified by $T = \left( \begin{array}{c} 1\, 1\\ 0\, 1 \end{array}\right)$ and the edges $P_1P_2$ and $P_2P_3$ are identified by $S = \left( \begin{array}{c} 0\, -1\\ 1\, 0 \end{array}\right)$. The elliptic fixed point $P_2$ of order 2 gives the relation $S^2=-1$ and the elliptic fixed point $P_1$ of order 3 gives the relation $(ST)^3=-1$. We now turn to the more complicated case of $\Gamma _0^*(25)$.

Fig. 3

A fundamental domain $\mathcal {F}$ of $\Gamma ^*_0(25)$ with three inequivalent parabolic vertices $P_0,P_2,P_4$ and three inequivalent elliptic vertices $P_1,P_3,P_5$ of order two

For $\Gamma _0^*(25)$ we find the fundamental domain $\mathcal {F}$ shown in Fig. 3. From the Ford circles bounding $\mathcal {F}$ one sees that one can choose the generators of $\Gamma ^*_0(25)$ as

$$\begin{aligned} T=\left( \begin{array}{cc} 1&{}1\\ 0 &{} 1\end{array}\right) , \ A=\left( \begin{array}{cc} 5&{}\frac{12}{5}\\ 10 &{} 5\end{array}\right) ,\ B=\left( \begin{array}{cc} 5&{}\frac{8}{5}\\ 15 &{} 5\end{array}\right) ,\ C=\left( \begin{array}{cc} 5&{}\frac{6}{5}\\ 20 &{} 5\end{array}\right) ,\ W=\left( \begin{array}{cc} 0&{}-\frac{1}{5}\\ 5 &{} 0\end{array}\right) . \end{aligned}$$

(A.40)

The relations between these generators are again obtained from considering the finite orbits of the vertices of $\mathcal {F}$ under the action of $\Gamma ^*_0(25)$. For example, A maps $P_1$ to $P_9$ and $T^{-1}$ maps $P_9$ back to to $P_1$. From similar considerations one concludes that one has three elliptic elements of order two (in $\Gamma ^*_0(25)/\{\pm 1\}$)

$$\begin{aligned} T^{-1}A, \ B^{-1}C,\ W, \qquad \textrm{fixing} \quad P_1=-\frac{1}{2}+\frac{i}{10}, \ P_3=-\frac{7}{25}+\frac{i}{25}, \ P_5=\frac{i}{5} , \end{aligned}$$

(A.41)

respectively. Analogously, we find the three inequivalent parabolic elements

$$\begin{aligned} T, \ A^{-1}B, \ C^{-1}W, \qquad \textrm{fixing} \qquad P_0=\infty , \ P_2=-\frac{2}{5}, \ P_4=-\frac{1}{5} , \end{aligned}$$

(A.42)

respectively.

Now, we set $k=4$ and give an explicit description of $H_{\text {par}}^1(\Gamma _0^*(25), V_2(\mathbb {Q}))$. For a cocycle $r \in Z_{\text {par}}^1(\Gamma _0^*(25), V_2(\mathbb {Q}))$ we write the five period polynomials corresponding to the generators $\gamma$ from (A.40) as

$$\begin{aligned} r(\gamma ) = a_\gamma ^{(2)} \tau ^2+ a_\gamma ^{(1)} \tau +a_\gamma ^{(0)} \end{aligned}$$

(A.43)

so that in total we have 15 rational coefficients $a_\gamma ^{(k)}$, $\gamma \in \{T,A,B,C,W \}$, $k=0,1,2$. The existence of the elliptic elements A.41 and the parabolic elements A.42 imposes six relations among them. For example, the cocycle relation yields $r(T^{-1} A)=r(A) - r(T)|_{-2}T^{-1}A$, but since $T^{-1}A$ is of order two one gets

$$\begin{aligned} 0 = r(T^{-1} A)|_{-2}T^{-1}A + r(T^{-1}A), \end{aligned}$$

(A.44)

yielding one relation between the coefficients of r(A) and r(T). For the representatives we can furthermore choose e.g. $r(T)=0$—two nontrivial conditions—as well as $r(W)=a_W^{(1)}\tau$—one nontrivial condition –, which leaves six independent parabolic cocycles, which we choose as in Table 5.

Table 5 Representatives for a basis of $H_{\text {par}}^1(\Gamma _0^*(25), V_2(\mathbb {Q}))$

We now work out the action of the Hecke operator $T_2$ as defined in (A.30) in detail. We choose $\sigma _1(2) = 3$ representatives for $\Gamma _0(25)\backslash \mathcal {M}_{2,25}$ as

$$\begin{aligned} \begin{aligned} M_1&= \begin{pmatrix} 2&{}0\\ 0 &{} 1 \end{pmatrix},&M_2&= \begin{pmatrix} 1&{}0\\ 0 &{} 2 \end{pmatrix},&M_3&= \begin{pmatrix} 1&{}1\\ 0 &{} 2 \end{pmatrix}. \end{aligned} \end{aligned}$$

(A.45)

In order to apply (A.30) we need to determine for $\gamma \in \Gamma _0^*(25)$ the permutations $\pi _\gamma$ as well as the $\gamma _i$ for $i=1,2,3$ as defined in (A.31) and express the latter as a word in terms of the chosen generators of $\Gamma _0^*(25)$, see Table 6.

Table 6 $\gamma _i$ and $\pi _\gamma (i)$ for $\gamma \in \{T,A,B,C,W\}$ as defined by $M_i\gamma =\gamma _i M_{\pi _\gamma (i)}$, where we decomposed $\gamma _i$ as a word in terms of the chosen generators of $\Gamma _0^*(25)$

Given this information we can compute the action of $T_2$ on the basis of $H_{\text {par}}^1(\Gamma _0^*(25), V_2(\mathbb {Q}))$ defined in Table 5 by repeatedly using the cocycle property. By (A.33), this action must have the same eigenvalues as the Hecke operator $T_2$ acting on $S_4(\Gamma ^*_0(25))$ and by the Eichler-Shimura isomorphism they must appear with multiplicity two. The calculation gives

$$\begin{aligned} \left( \begin{array}{c} {[}r_1{]}|_{-2}\, T_2 \\ \vdots \\ {[}r_6{]}|_{-2}\, T_2 \end{array}\right) = \left( \begin{array}{cccccc} -4 &{} -\frac{6}{5} &{} -\frac{9}{5} &{} -\frac{4}{5} &{} -\frac{8}{5} &{} 0 \\ -24 &{} \frac{184}{5} &{} \frac{246}{5} &{} \frac{336}{5} &{} \frac{672}{5} &{} -12 \\ 16 &{} -\frac{106}{5} &{} -\frac{139}{5} &{} -\frac{204}{5} &{} -\frac{408}{5} &{} 8 \\ 104 &{} \frac{366}{5} &{} \frac{539}{5} &{} \frac{384}{5} &{} \frac{798}{5} &{} -8 \\ -42 &{} -\frac{178}{5} &{} -\frac{262}{5} &{} -\frac{192}{5} &{} -\frac{399}{5} &{} 4 \\ 74 &{} \frac{122}{5} &{} \frac{182}{5} &{} \frac{96}{5} &{} \frac{199}{5} &{} 0 \end{array} \right) \left( \begin{array}{c} {[}r_1{]} \\ \vdots \\ {[}r_6{]} \end{array}\right) \end{aligned}$$

(A.46)

and so the eigenvalues are $\{1,1,4,4,-4,-4\}$. Analogously, we compute the matrix associated with the involution $\varepsilon$. For any generator $\gamma \in \{ T,A,B,C,W\}$ one has $\epsilon \gamma \epsilon = \gamma ^{-1}$ and one then obtains

$$\begin{aligned} \left( \begin{array}{c} {[}r_1{]}|_{-2}\, \epsilon \\ \vdots \\ {[}r_6{]}|_{-2}\, \epsilon \end{array}\right) = \left( \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 12 &{} 7 &{} 12 &{} 0 &{} 0 &{} 0 \\ -8 &{} -4 &{} -7 &{} 0 &{} 0 &{} 0 \\ 8 &{} 12 &{} 18 &{} 9 &{} 20 &{} 0 \\ -4 &{} -6 &{} -9 &{} -4 &{} -9 &{} 0 \\ 4 &{} 6 &{} 9 &{} 8 &{} 16 &{} -1 \\ \end{array} \right) \left( \begin{array}{c} {[}r_1{]} \\ \vdots \\ {[}r_6{]} \end{array}\right) . \end{aligned}$$

(A.47)

We can now choose an eigenbasis of $H^1_{\textrm{par}}(\Gamma _0^*(N),V_2(\mathbb {Q}))$ with respect to the action of $T_2$ and $\varepsilon$. A possible choice is given in Table 7. The lower index of $r_\lambda ^\pm$ indicates the eigenvalue $\lambda$ of the associated cohomology class with respect to $T_2$ and the upper index denotes the eigenvalue $\pm 1$ of the associated cohomology class with respect to the involution $\varepsilon$.

Table 7 Representatives for an eigenbasis of $H_{\text {par}}^1(\Gamma _0^*(25), V_2(\mathbb {Q}))$ w.r.t. $T_2$ and $\varepsilon$

1.2 Meromorphic cusp forms and quasiperiods

In the previous section we have seen that there is an isomorphism

$$\begin{aligned} H^1_{\textrm{par}}(\Gamma ,V_{k-2}(\mathbb {C})) \, \cong \, S_k(\Gamma ) \oplus \overline{S_k(\Gamma )} . \end{aligned}$$

(A.48)

For the case $k=2$ this corresponds to the usual Hodge decomposition $H^1 = H^{1,0} \oplus H^{0,1}$ for complex curves and the complex conjugation makes this decomposition non-algebraic. In this case the algebraic version of $H^1$ can be realized by holomorphic differentials (differentials of the first kind) and meromorphic differentials with vanishing residues (differentials of the second kind). The integration of these forms gives a well defined pairing with the homology and taking the quotient by derivatives of meromorphic forms one obtains a space that is isomorphic to $H^1$ and defined algebraically. Instead of the Hodge decomposition we then have a filtration into classes that can be represented by differentials of the first and second kind, respectively. In this section we discuss the algebraic analogue of the isomorphism (A.48) by considering meromorphic modular forms. This will allow us to define quasiperiods as the periods of certain meromorphic modular forms. The theory of meromorphic cusp forms and their associated period polynomials was first introduced by Eichler [86] and later independently rediscovered by Brown [89] and one of the authors in the context of [90].

1.2.1 Meromorphic cusp forms and their period polynomials

We want to extend the period map $r: S_k(\Gamma ) \rightarrow H^1_{\text {par}}(\Gamma ,V_{k-2}(\mathbb {C}))$ to the space of meromorphic modular forms

$$\begin{aligned} M_k^{\textrm{mero}}(\Gamma ) = \{F:\overline{\mathfrak {H}}\rightarrow \mathbb {P}^1(\mathbb {C}) \mid F \ \textrm{meromorphic} \text { and } F|_{k}\gamma = F \ \forall \gamma \in \Gamma \} . \end{aligned}$$

(A.49)

However, to have an Eichler integral, we need to restrict to forms that are $(k-1)$-st derivatives. By simple connectivity it is enough to require that they are locally $(k-1)$-st derivatives and we thus define

$$\begin{aligned} S_k^{\textrm{mero}}(\Gamma ) = \{ F \in M_k^{\textrm{mero}}(\Gamma ) \mid F \text { is locally a (k-1)-st derivative} \} . \end{aligned}$$

(A.50)

Concretely, this means that for each $\tau _0 \in \mathfrak {H}$ the coefficients of $(\tau -\tau _0)^m$ in the Laurent expansion around $\tau _0$ vanish for $m=-1,...,-(k-1)$ and that for each cusp the constant coefficient in the Fourier expansion vanishes. For any $F \in S_k^{\textrm{mero}}(\Gamma )$ one can then choose an Eichler integral $\widetilde{F}$, i.e. a meromorphic modular form such that $D^{k-1}\widetilde{F}=F$, and compute the period polynomials $r_F(\gamma ) = \widetilde{F}|_{2-k}(\gamma -1)(\tau )$ for $\gamma \in \Gamma$. These are polynomials by Bol’s identity and as in the case of holomorphic cusp forms one finds that they are parabolic cocycles and induce a well defined class $[r_F] \in H^1_{\text {par}}(\Gamma ,V_{k-2}(\mathbb {C}))$ which does not depend on the choice of Eichler integral. Bol’s identity also implies that $D^{k-1}M_{2-k}^{\textrm{mero}}(\Gamma ) \subseteq S_k^{\textrm{mero}}(\Gamma )$ and of course the classes in $H^1(\Gamma ,V_{k-2}(\mathbb {C}))$ associated with elements in $D^{k-1}M_{2-k}^{\textrm{mero}}(\Gamma )$ are trivial. This motivates introducing the quotient

$$\begin{aligned} \mathbb {S}_k(\Gamma ) = S_k^{\textrm{mero}}(\Gamma ) / (D^{k-1}M_{2-k}^{\textrm{mero}}(\Gamma )) . \end{aligned}$$

(A.51)

Note that the Riemann-Roch theorem implies that one can choose the representatives to have poles only in an arbitrary non-zero subset of $\overline{\mathfrak {H}}$ closed under the action of $\Gamma$, for instance the set of all cusps (if there are cusps) or the set of cusps equivalent to $\infty$ (if $\infty$ is a cusp). For suitable $\Gamma$ we therefore have canonical isomorphisms

$$\begin{aligned} \mathbb {S}_k^{[\infty ]}(\Gamma ) \; \cong \; \mathbb {S}_k^{!}(\Gamma )\; \cong \; \mathbb {S}_k(\Gamma ) , \end{aligned}$$

(A.52)

where the first two spaces are defined as in (A.51) but restricting to forms with possible poles only at $[\infty ]$ or only at the cusps, respectively.

In the following we explain that the period map gives an isomorphism between $\mathbb {S}_k(\Gamma )$ and $H^1_{\text {par}}(\Gamma ,V_{k-2}(\mathbb {C}))$. We start by defining a useful pairing.

Proposition 6

(Eichler pairing). There is a pairing $\{\;,\;\}: S_k^{\textrm{mero}}(\Gamma )\times S_k^{\textrm{mero}}(\Gamma )\rightarrow \mathbb {C}$ defined by

$$\begin{aligned} \{F,G\} = (2\pi i)^k \sum _{\tau \in \Gamma \backslash \overline{\mathfrak {H}}} {\text {Res}}_\tau ({\widetilde{F}} G \, \hbox {d}\tau ) . \end{aligned}$$

(A.53)

This pairing is $(-1)^{k+1}$–symmetric and descends to $\mathbb {S}_k(\Gamma )\times \mathbb {S}_k(\Gamma )$.

Proof

First note that the right-hand side of (A.53) makes sense because the sum is finite (only finitely many orbits have poles) and the individual residues do not depend on the choice of $\tau$ in the $\Gamma$-orbit (the difference of the residues at $\tau$ and $\gamma \tau$ is $r_F(\gamma )G \, \hbox {d}\tau$ which cannot have any residues since G is a $(k-1)$-st derivative and $r_F(\gamma )$ is a polynomial of degree at most $k-2$). Similarly, the pairing does not depend on the choice of Eichler integral since $\widetilde{F}$ is unique up to a polynomial p of degree $k-2$ and $p\, G\, \hbox {d}\tau$ again has no residues. The $(-1)^{k+1}$–symmetry follows since $\widetilde{F}G-(-1)^{k+1}F\widetilde{G}$ is a derivative. Because of this symmetry it just remains to prove that $\{F,G\}$ vanishes for any $F \in D^{k-1}M_{2-k}^{\textrm{mero}}(\Gamma )$. This is clear since one can choose $\widetilde{F}$ to be in $M_{2-k}^{\textrm{mero}}(\Gamma )$ and then $\widetilde{F}G \, \hbox {d}\tau$ is a well defined meromorphic differential on the compact curve $\Gamma \backslash \overline{\mathfrak {H}}$ and hence the sum of its residues vanishes. $\square$

Theorem 2

(Eichler). The natural map $S_k(\Gamma ) \rightarrow \mathbb {S}_k(\Gamma )$ induced by inclusion and the map $F \mapsto \{F, \ \}$ give the short exact sequence

(A.54)

where denotes the dual space of $S_k(\Gamma )$.

Proof

The first (non-trivial) map is injective since the period polynomial of a holomorphic cusp form determines the form uniquely. The composite of the first two maps is trivial since holomorphic functions don’t have poles. Eichler [86] shows that the kernel of the second map is exactly the image of the first map and that the second map is surjective. $\square$

This theorem implies that $\mathbb {S}_k(\Gamma )$ is (non-canonically) isomorphic to . Hence the domain and the codomain of the period map $r: \mathbb {S}_k(\Gamma ) \rightarrow H^1_{\textrm{par}}(\Gamma ,V_{k-2}(\mathbb {C}))$ have the same dimension and since the map is injective it gives an isomorphism

$$\begin{aligned} \mathbb {S}_k(\Gamma ) \, \cong \, H^1_\textrm{par}(\Gamma ,V_{k-2}(\mathbb {C})) . \end{aligned}$$

(A.55)

We now restrict to $\Gamma = \Gamma _0(N)$ to introduce Hecke operators for meromorphic modular forms. For holomorphic modular forms we defined these in (A.7) and we use the same definition for meromorphic modular forms. By Bol’s identity it follows that they also descend to $\mathbb {S}_k(\Gamma _0(N))$ and we have the following proposition.

Proposition 7

The pairing $\{\;, \; \}$ is equivariant with respect to the Hecke operators.

Proof

Without loss of generality we can restrict to meromorphic modular forms F and G that only have poles at cusps equivalent to $\infty$. In terms of the Fourier coefficients $a_m$ and $b_m$ of F and G, respectively, we choose the Eichler integrals

$$\begin{aligned} \widetilde{F}(\tau ) = \sum _{m \ne 0 \atop m\gg -\infty }\frac{a_m}{m^{k-1}} q^m \qquad \text {and} \qquad \widetilde{G}(\tau ) = \sum _{m \ne 0 \atop m\gg -\infty }\frac{b_m}{m^{k-1}} q^m . \end{aligned}$$

(A.56)

This gives

$$\begin{aligned} \{F,G|_{k}T_n \}&= (2\pi i)^{k-1}\sum _{m \ne 0 \atop -\infty \ll m\ll \infty } \sum _{r | (m,n) \atop r>0} \frac{a_{-m}}{(-m)^{k-1}} r^{k-1}b_{mn/r^2} \nonumber \\&= (-1)^{k-1}(2\pi i)^{k-1}\sum _{m' \ne 0 \atop -\infty \ll m'\ll \infty } \sum _{r' | (m',n) \atop r'>0} r'^{k-1}a_{m'n/r'^2} \frac{b_{-m'}}{(-m')^{k-1}} \nonumber \\&=(-1)^{k-1} \{G,F|_{k}T_n \} \nonumber \\&=\{F|_{k}T_n ,G\} , \end{aligned}$$

(A.57)

where $m'=-mn/r^2$ and $r'=n/r$. $\square$

Just as in the case of holomorphic cusp forms, the period map $r: \mathbb {S}_k(\Gamma ) \rightarrow H^1_\textrm{par}(\Gamma ,V_{k-2}(\mathbb {C}))$ is compatible with the action of the Hecke operators. The proposition above further shows that the (non-canonical) isomorphism between $\mathbb {S}_k(\Gamma )$ and is also compatible with the action of the Hecke operators.

The above considerations show that associated with any newform $f \in S_k(\Gamma _0(N))$ we have a 2-dimensional subspace of $\mathbb {S}_k(\Gamma _0(N))$ with the same Hecke eigenvalues. Let $F\in S^{\text {mer}}_k(\Gamma _0(N))$ be such that [f] and [F] generate this subspace. We can choose F to have poles only at cusps equivalent to $\infty$ and Fourier coefficients in $\mathbb {Q}(f)$, and then call F (or [F]) a meromorphic partner of f. In Proposition 5 we showed that

$$\begin{aligned} {[}r_f] = \omega ^+_f [r^+] + \omega ^-_f [r^-] \end{aligned}$$

(A.58)

for $r^\pm \in Z_{\text {par}}^1(\Gamma _0(N),V_{k-2}(\mathbb {Q}(f)))$ and used this to define the periods $\omega ^\pm _f$, which are unique up to multiplication by $\mathbb {Q}(f)$. Completely analogously we have

$$\begin{aligned} {[}r_F] = \eta ^+_F [r^+] + \eta ^-_F [r^-] \end{aligned}$$

(A.59)

for the same $r^\pm$, which defines the quasiperiods $\eta ^\pm _F$. Note that these only depend on the class of F. We finish this section by giving a quadratic relation fulfilled by the periods and quasiperiods.

Proposition 8

(“Legendre Relation”). Let f be a newform with meromorphic partner F. Then the associated periods and quasiperiods satisfy

$$\begin{aligned} (\omega ^+_f\eta ^-_F-\omega ^-_f\eta ^+_F) \, \in \, (2\pi i)^{k-1}\mathbb {Q}(f) . \end{aligned}$$

(A.60)

Proof

First note that clearly $\{f,F\} \in (2\pi i)^{k-1}\mathbb {Q}(f)$. The idea now is to relate the pairing $\{ \cdot , \cdot \}$ to a pairing on $H^1_{\text {par}}(\Gamma _0(N),V_{k-2}(\mathbb {C}))$. We give an explicit proof for level 1 which goes along the lines of similar calculations in [91] and [92]. $\text {SL}(2,\mathbb {Z})$ is generated by T and $S=\left( \begin{array}{c} 0\, -1\\ 1\,0 \end{array}\right)$ satisfying $S^2 = (ST)^3 = -1$ and a standard (non-strict) fundamental domain is given by . In the following we abuse the notation and denote by F also a representative of F without poles on the boundary of $\mathcal {F}$. We then have

$$\begin{aligned} \{ f,F \}&= (2\pi i)^{k-1} \int _{\partial \mathcal {F}} \widetilde{f}F \, \hbox {d}\tau \end{aligned}$$

(A.61)

$$\begin{aligned}&= (2\pi i)^{k-1} \int _{\frac{i\sqrt{3}-1}{2}}^\infty (\widetilde{f}|_{2-k}(T-1))F \, \hbox {d}\tau \end{aligned}$$

(A.62)

$$\begin{aligned}&\quad +(2\pi i)^{k-1}\int _i^{\frac{i\sqrt{3}-1}{2}}(\widetilde{f}|_{2-k}(S-1))F \, \hbox {d}\tau . \end{aligned}$$

(A.63)

For $\tau _0 = \frac{i\sqrt{3}+1}{2}$ we have $T^{-1}\tau _0 = S^{-1}\tau _0$ and with the choice $\widetilde{f} = \widetilde{f}_{\tau _0}$ this gives $r_{f,\tau _0}(S) = r_{f,\tau _0}(T)$ and thus

$$\begin{aligned} \{ f,F \} = (2\pi i)^{k-1} \int _i^\infty r_{f,\tau _0}(T)F \, \hbox {d}\tau . \end{aligned}$$

(A.64)

From $S^2=-1$ we further get $r_{f,\tau _0}(S)|_{2-k}S = -r_{f,\tau _0}(S)$ and so

$$\begin{aligned} \{ f,F \}&= \frac{1}{2}(2\pi i)^{k-1} \int _0^\infty r_{f,\tau _0}(T)F \, \hbox {d}\tau \end{aligned}$$

(A.65)

$$\begin{aligned}&= -\frac{(k-2)!}{2}\sum _{i=0}^{k-2} (-1)^i{k-2 \atopwithdelims ()i}^{-1} r_{f,\tau _0}(T)_i r_{F,\infty }(S)_{k-2-i} \end{aligned}$$

(A.66)

$$\begin{aligned}&=: -\frac{(k-2)!}{2} <r_{f,\tau _0}(T),r_{F,\infty }(S)> . \end{aligned}$$

(A.67)

Here $p_i$ denotes the coefficient of $\tau ^i$ for $p \in V_{k-2}(\mathbb {C})$ and it is straightforward to show that the defined pairing $< \cdot , \cdot >: V_{k-2}(\mathbb {C}) \times V_{k-2}(\mathbb {C}) \rightarrow \mathbb {C}$ is ${\textrm{SL}}(2,\mathbb {Z})$ invariant. We now want to replace $r_{f,\tau _0}$ by $r_{f,\infty }$. Using the T invariance of $\widetilde{f}_{\infty }$, the ${\textrm{SL}}(2,\mathbb {Z})$ invariance of the pairing $< \cdot , \cdot>$ and the cocycle relations associated with the identities $S^2 = (ST)^3 = -1$ gives

$$\begin{aligned}<r_{f,\tau _0}(T),r_{F,\infty }(S)>&= <(\widetilde{f}_{\tau _0}-\widetilde{f}_{\infty })|_{2-k}(T-1),r_{F,\infty }(S)>\end{aligned}$$

(A.68)

$$\begin{aligned}&= <\widetilde{f}_{\tau _0}-\widetilde{f}_{\infty },r_{F,\infty }(S)|_{2-k}(T^{-1}-1)> \end{aligned}$$

(A.69)

$$\begin{aligned}&= -<\widetilde{f}_{\tau _0}-\widetilde{f}_{\infty },r_{F,\infty }(S)|_{2-k}(ST^{-1}+1)> \end{aligned}$$

(A.70)

$$\begin{aligned}&= -<\widetilde{f}_{\tau _0}-\widetilde{f}_{\infty },r_{F,\infty }(S)|_{2-k}((TS)^2+1)> \end{aligned}$$

(A.71)

$$\begin{aligned}&= -\frac{1}{3}<\widetilde{f}_{\tau _0}-\widetilde{f}_{\infty },r_{F,\infty }(S)|_{2-k}((TS)^2+1-2TS)> \end{aligned}$$

(A.72)

$$\begin{aligned}&= -\frac{1}{3}<\widetilde{f}_{\tau _0}-\widetilde{f}_{\infty },r_{F,\infty }(S)|_{2-k}(TST-T)(S-T^{-1})> \end{aligned}$$

(A.73)

$$\begin{aligned}&= \frac{1}{3}<(\widetilde{f}_{\tau _0}-\widetilde{f}_{\infty })|_{2-k}(T-S),r_{F,\infty }(S)|_{2-k}(TST-T)> \end{aligned}$$

(A.74)

$$\begin{aligned}&= \frac{1}{3}<r_{f,\infty }(S),r_{F,\infty }(S)|_{2-k}(ST^{-1}S-T)> \end{aligned}$$

(A.75)

$$\begin{aligned}&= \frac{1}{3}<r_{f,\infty }(S)|_{2-k}(T-T^{-1}),r_{F,\infty }(S)>. \end{aligned}$$

(A.76)

We note that any coboundary which vanishes on T comes from a constant polynomial and hence this expression is invariant under shifting the parabolic cocycles by such coboundaries. In particular, we can define a pairing $< \cdot , \cdot>$ on $H^1_{\text {par}}({\textrm{SL}}(2,\mathbb {Z}),V_{k-2}(K))$ by

$$\begin{aligned}<[r_1],[r_2]> = -\frac{(k-2)!}{6} <r_1(S)|_{2-k}(T-T^{-1}),r_2(S)> , \end{aligned}$$

(A.77)

where $r_1,r_2$ must be chosen such that $r_1(T)=r_2(T)=0$. We see that this pairing is $\varepsilon$ invariant and we conclude that

$$\begin{aligned} \{ f,F \} = (\omega ^+_f\eta ^-_F-\omega ^-_f\eta ^+_F) \underbrace{<r^+,r^->}_{\in \, \mathbb {Q}(f)} \, \in \, (2\pi i)^{k-1}\mathbb {Q}(f) \end{aligned}$$

(A.78)

which finishes the proof for $\textrm{SL}(2,\mathbb {Z})$. The proof for higher levels can be done by using Shapiro’s lemma [93] or in a way similar to the calculations in [94]. $\square$

1.2.2 Computation of $\mathbb {S}_4 (\Gamma _0(25))$

In this subsection we explain the explicit computation of a Hecke eigenbasis of $\mathbb {S}_4(\Gamma _0(N))$. We do this in detail for the case of $\Gamma _0(25)$, the general case being similar. To this end, we first discuss the construction of weakly holomorphic modular forms with a given pole order at $\infty$, use these to construct a basis of $\mathbb {S}_4(\Gamma _0(25))$ and diagonalize the action of the Hecke algebra. Since newforms are also eigenforms under Atkin–Lehner involutions and since there are no old forms of level 25 and weight 4 this also allows us to write down a Hecke eigenbasis of $\mathbb {S}_4(\Gamma ^*_0(25))$.

For any $\Gamma$, one can give $X(\Gamma ) = \Gamma \backslash \overline{\mathfrak {H}}$ the structure of a Riemann surface, and although non-zero $F\in M_k^{\textrm{mero}}(\Gamma )$ with $k \ne 0$ are not well-defined on $X(\Gamma )$, one can still define a vanishing order $\text {ord}_\tau (F)$ at any $\tau \in X(\Gamma )$. E.g. if $T \in \Gamma$ and $\Gamma _\infty = <T>$, then the vanishing order at the cusp $\infty$ is given by the lowest exponent in the Fourier expansion around $\infty$. The Eichler–Selberg trace formula or the Riemann–Roch formula imply that the total order of vanishing of any non-zero $g \in M_k(\Gamma )$ is given by $\kappa _\Gamma k$, where $\kappa _{\Gamma } = \frac{1}{4\pi }{\textrm{Vol}}(\Gamma \backslash \mathfrak {H})$ in terms of the hyperbolic volume. Since $[{\textrm{SL}}(2,\mathbb {Z}):\Gamma _0(N)] = N \prod _{p|N}(1+\frac{1}{p})$, we have $\kappa _{\Gamma _0(N)} = \frac{N}{12} \prod _{p|N}(1+\frac{1}{p})$ and $\kappa _{\Gamma ^*_0(N)} = \frac{1}{2^e}\kappa _{\Gamma _0(N)}$, where e is the number of prime factors of N. Restricting to the case $T \in \Gamma$ we now set $M_k^{[\infty ,M]}(\Gamma ) = \{ f \in M_k^{[\infty ]}(\Gamma ) \mid {\textrm{ord}}_\infty f \ge M \}$, where $M_k^{[\infty ]}(\Gamma )$ consists of meromorphic modular forms with poles only at cusps equivalent to $\infty$, and denote by $S_k^{[\infty ,M]}(\Gamma )$ the subspace with vanishing residues. Let $h \in M_a(\Gamma )$ have the maximal order of vanishing $A=\kappa _\Gamma a$ at $\infty$ (which exists for a large enough). For $\ell$ large enough we have the short exact sequence

$$\begin{aligned} 0 \longrightarrow M_{2-k}^{[\infty ,-\ell A+1]}(\Gamma ) \xrightarrow []{D^{k-1}} S_k^{[\infty ,-\ell A+1]}(\Gamma ) \longrightarrow \mathbb {S}_k(\Gamma ) \longrightarrow 0 . \end{aligned}$$

(A.79)

The multiplication by $h^\ell$ gives an isomorphism between $M_{2-k}^{[\infty ,-\ell A+1]}(\Gamma )$ and a subspace of $M_{2-k+\ell A}(\Gamma )$ (of codimension at most 1) and also an isomorphism between $S_{2-k}^{[\infty ,-\ell A+1]}(\Gamma )$ and a subspace of $S_{k+\ell A}(\Gamma )$ (of codimension at most 1). Hence the construction of $\mathbb {S}_k(\Gamma )$ can be reduced to linear algebra in these finite dimensional spaces. We now specialize to the case $\Gamma = \Gamma _0(N)$. As explained in [95], the form h necessarily can be realized as an eta quotient,

$$\begin{aligned} h(\tau ) = \prod _{m\mid N} \eta (m\tau )^{r_m}, \qquad r_m \in \mathbb {Z}\ , \end{aligned}$$

(A.80)

and we have the following expressions for the weight k and for the vanishing order at $\infty$, respectively:

$$\begin{aligned} \begin{aligned} k&= \frac{1}{2}\sum _{m\mid N} r_m,&\text {ord}_\infty (h)&= \frac{1}{24}\sum _{m\mid N} mr_m . \end{aligned} \end{aligned}$$

(A.81)

For cusps of the form $a/c \in \mathbb {P}^1(\mathbb {Q})$ with $\gcd (a,c)= 1$, c|N and $c > 0$, the order $\text {ord}_{a/c}(h)$ evaluates to

$$\begin{aligned} \text {ord}_{a/c}(h) = \frac{N}{\gcd (c,N/c)c} \frac{1}{24} \sum _{m \mid N} \frac{\gcd (m,c)^2r_m}{m} . \end{aligned}$$

(A.82)

We now consider the case $N=25$. Since we are interested in eigenforms, we will consider both $\Gamma _0(25)$ and its extension $\Gamma _0^*(25)$ by $W = \left( \begin{array}{c} 0\ -1/5 \\ 5 \ 0 \end{array}\right)$. A fundamental domain for the latter was constructed in Sect. A.1.4, and to get a fundamental domain of $\Gamma _0(25)$ we can take the union of that domain and (any $\Gamma _0(25)$ translation of) its image under W. One finds that there are six inequivalent cusps at $\infty ,0,\frac{1}{5}, \frac{2}{5},\frac{3}{5},\frac{4}{5}$, and two inequivalent elliptic fixed points of order 2 at $P_{\pm }=\frac{1}{25}(i\pm 7)$ (fixed by $\left( \begin{array}{c} 7\;\;\,-2\\ 25\;-7 \end{array}\right)$). Since the genus of $X_0(25)$ is zero, we can construct the weakly holomorphic modular forms from a Hauptmodul $\phi$ of $\Gamma _0(25)$, i.e. a generator of the field of meromorphic modular functions $M^{\textrm{mero}}_0(\Gamma _0(25)) = \mathbb {C}(\phi )$. We take $\phi$ to be

$$\begin{aligned} \phi (\tau ) = \frac{\eta (\tau )}{\eta (25 \tau )}=\frac{1}{q}-1-q+q^4+q^6-q^{11}+\cdots . \end{aligned}$$

(A.83)

The function $\phi$ has a single pole at $\infty$ and vanishes at 0 to first order. We also need the unique normalized form $h\in M_4(\Gamma _0(25))$ with the maximal vanishing order $\frac{4}{12}[{\textrm{SL}}(2,\mathbb {Z}):\Gamma _0(25)] = 10$ at $\infty$. This is given by the eta product

$$\begin{aligned} h(\tau ) = \frac{\eta (25 \tau )^{10}}{\eta (5 \tau )^2}=q^{10}+2 q^{15}+5 q^{20}+\cdots . \end{aligned}$$

(A.84)

For the construction of the meromorphic modular forms we also introduce $\delta \in S_4(\Gamma _0(25))$ defined by

$$\begin{aligned} \delta (\tau ) = \eta (5 \tau )^4 \eta ( 25 \tau )^4=q^5-4 q^{10}+2 q^{15}+8 q^{20}-5 q^{25}+\cdots , \end{aligned}$$

(A.85)

and the Eisenstein series $e \in M_2(\Gamma _0(25))$ defined by

$$\begin{aligned} e(\tau ) = \frac{\eta (25\tau )^5}{\eta (5\tau )} \sqrt{\phi (\tau )^2+2 \phi (\tau )+ 5}=\frac{1}{5} \sum _{\begin{array}{c} a,b>0\\ a+b\equiv 0 {\textrm{mod}}5\\ a\not \equiv 0{\textrm{mod}}5 \end{array}}a q^{a b} = q^4 + q^6 + \cdots . \end{aligned}$$

(A.86)

By (A.82), $\delta$ vanishes to order 5, 1, 1, 1, 1, 1 at the six cusps and so does not have any other zeros. e vanishes with order 4 at $\infty$ and with order 1/2 at the elliptic fixed points $P_\pm$ and so does not have any other zeros. Hence $M_2^{[\infty ]}(\Gamma _0(25))=e \mathbb {C}[\phi ]$ and moreover

$$\begin{aligned} M^{[\infty ]}_{-2}(\Gamma _0(25)) = \frac{e}{h}\mathbb {C}[\phi ]. \end{aligned}$$

(A.87)

It follows that non-zero elements in $M^{[\infty ]}_{-2}(\Gamma _0(25))$ have vanishing order at most $-6$ at $\infty$ and since D does not change the order at $\infty$, we can construct representatives of a basis of $S_4^{[\infty ]}(\Gamma _0(25))/ D^3 M^{[\infty ]}_{-2}(\Gamma _0(25))$ with vanishing order $-5\le m \le 5$ and $m \ne 0$ at $\infty$. It follows that possible representatives are given by forms $F_i = \delta \, p_i(\phi )$, where $p_1,...,p_{10}$ are linearly independent polynomials of degree at most 10 chosen such that the forms $F_i$ have no constant coefficients.

Using (A.9) to compute the action of the Hecke operator $T_2$ on the constructed basis one finds that $T_2$ has the eigenvalue $-4$ with multiplicity 4 and the eigenvalues $-1$, 1 and 4 with multiplicity 2. To define a Hecke eigenbasis, we further split these by considering the Atkin-Lehner involution W. To do this we construct representatives that are eigenforms under this involution. To this end we note that W acts as

$$\begin{aligned} h|_4 {W} = \frac{1}{5^5} h \phi ^{10},\quad \delta |_4 {W} = \frac{1}{5^2} \delta \phi ^4 ,\quad \phi |_0 {W} = \frac{5}{\phi } ,\quad e|_2 {W} = -\frac{1}{5^2} e \phi ^4 . \end{aligned}$$

(A.88)

It is straightforward to construct basis elements with definite eigenvalue under the action of W. In particular, the invariant combination

$$\begin{aligned} \phi _+ = \phi + \frac{5}{\phi } = q^{-1}-1+4q+5q^{2}+10q^{3}+\cdots \ \end{aligned}$$

(A.89)

is a Hauptmodul of $\Gamma ^*_0(25)$ and the unique normalized form $h_+ \in M_4(\Gamma ^*_0(25))$ with the maximal vanishing order 5 at $\infty$ is given by

$$\begin{aligned} h_+ = h \phi ^5 = q^{5}-5q^{6}+5q^{7}+10q^{8}-15q^{9} +\cdots . \end{aligned}$$

(A.90)

Similarly, the unique normalized form $e_+ \in M_2(\Gamma ^*_0(25))$ with vanishing order 1 at $\infty$ is given by

$$\begin{aligned} e_+ = e\, (\phi ^3-5\phi ) = q-3q^{2}-4q^{3}+7q^{4}+12q^{6}+\cdots \ . \end{aligned}$$

(A.91)

This is unique since any non-zero element of $M_2(\Gamma ^*_0(25))$ must vanish at least to order 1/2 at the three inequivalent elliptic fixed points from Fig. 3. This shows that ${M_2^{[\infty ]}(\Gamma ^*_0(25)) = e_+\mathbb {C}[\phi _+]}$ and

$$\begin{aligned} M_{-2}^{[\infty ]}(\Gamma ^*_0(25)) = \frac{e_+}{h_+}\mathbb {C}[\phi _+] . \end{aligned}$$

(A.92)

To complete the analysis we also introduce the form $h_- \in M_4(\Gamma _0(25))$ defined by

$$\begin{aligned} h_- = h \, (\phi ^6 - 5\phi ^4) = q^4 - 6 q^5 + 4 q^6 + 30 q^7 - 40 q^8 - 38 q^9+\cdots , \end{aligned}$$

(A.93)

which is anti-invariant under W and vanishes to order 4 at $\infty$. This is the unique normalized form with these properties since any anti-invariant form from $M_4(\Gamma _0(25))$ also has to vanish at the fixed point i/5 of W. Using the forms $\delta _\pm \in S_4(\Gamma _0(25))$ defined by

$$\begin{aligned} \delta _+&= \delta \phi ^2 = q^3 - 2 q^4 - q^5 + 2 q^6 + q^7 - 2 q^8+\ldots \nonumber \\ \delta _-&= \delta \, (\phi ^3-5\phi ) = q^2 - 3 q^3 - 5 q^4 + 10 q^5 + 5 q^6 - 4 q^7+\cdots \end{aligned}$$

(A.94)

we can now construct a basis of invariant forms $F_{+,i} = \delta _+ \, p_{+,i}(\phi _+)$, where $p_{+,1},...,p_{+,6}$ are linearly independent polynomials of degree at most 6 with the property that the forms $F_{+,i}$ do not have constant coefficients. A basis of anti-invariant forms is given by $F_{-,i} = \delta _- \, p_{-,i}(\phi _+)$, where $p_{-,1},...,p_{-,4}$ are linearly independent polynomials of degree at most 4 with the property that the forms $F_{-,i}$ do not have constant coefficients. We diagonalize the action of the Hecke operator $T_2$ on this basis and conclude that representatives for a Hecke eigenbasis of $\mathbb {S}_4(\Gamma _0(25))$ are given by

$$\begin{aligned} f_{+,-4}&= \delta _+ \, (\phi _+^2-10) = q - 4q^2 + 2q^3 + 8q^4 + 20q^5 - 8q^6 + 6q^7 - 23q^9+\cdots \nonumber \\ F_{+,-4}&= \frac{\delta _+}{27}(27\phi _+^6+240\phi _+^5+320\phi _+^4-2580\phi _+^3-9385\phi _+^2-9900\phi _+-1900)\nonumber \\&= q^{-3} + \frac{8}{9}q^{-2} - \frac{10}{27}q^{-1} + \frac{1100}{27}q^2 + \frac{6586}{27}q^3 + \frac{31760}{27}q^4 + \frac{40475}{9}q^5+\cdots \nonumber \\ f_{+,1}&= \delta _+ \, (\phi _+^2+ 5 \phi _+ + 10) = q + q^2 + 7q^3 - 7q^4 + 7q^6 + 6q^7 - 15q^8 + 22q^9+\cdots \nonumber \\ F_{+,1}&= \frac{\delta _+ }{27}(27\phi _+^6+220\phi _+^5+190\phi _+^4-2580\phi _+^3-7975\phi _+^2-7275\phi _+-1250)\nonumber \\&= q^{-3} + \frac{4}{27}q^{-2} + \frac{665}{27}q^2 + \frac{5141}{27}q^3 + \frac{8875}{9}q^4 + \frac{34375}{9}q^5+\cdots \nonumber \\ f_{+,4}&= \delta _+ \, (\phi _+^2+ 8 \phi _+ + 10) = q + 4q^2 - 2q^3 + 8q^4 - 8q^6 - 6q^7 - 23q^9+\cdots \nonumber \\ F_{+,4}&= \frac{\delta _+}{27}(27\phi _+^6+208\phi _+^5+100\phi _+^4-2652\phi _+^3-7141\phi _+^2-5148\phi _++340)\nonumber \\&= q^{-3} - \frac{8}{27}q^{-2} - \frac{2}{9}q^{-1} + \frac{404}{27}q^2 + \frac{4274}{27}q^3 + \frac{2384}{3}q^4 + \frac{29375}{9}q^5+\cdots \nonumber \\ f_{-,-4}&= \delta _- \phi _+ = q - 4q^2 + 2q^3 + 8q^4 - 30q^5 - 8q^6 + 6q^7 - 23q^9+\cdots \nonumber \\ F_{-,-4}&= \frac{\delta _-}{8}(8\phi _+^4+57\phi _+^3+110\phi _+^2+11\phi _+-100) \nonumber \\&= q^{-2} + \frac{1}{8}q^{-1} - \frac{73}{4}q^2 - \frac{331}{4}q^3 - 347q^4 - \frac{9355}{8}q^5+\cdots \nonumber \\ f_{-,-1}&= \delta _- \, (\phi _++3) = q - q^2 - 7q^3 - 7q^4 + 7q^6 - 6q^7 + 15q^8 + 22q^9+\cdots \nonumber \\ F_{-,-1}&= \frac{\delta _-}{4}(4\phi _+^4+30\phi _+^3+64\phi _+^2+22\phi _+-29) \nonumber \\&= q^{-2} + \frac{1}{2}q^{-1} - \frac{73}{4}q^2 - \frac{421}{4}q^3 - \frac{1607}{4}q^4 - 1250q^5+\cdots , \end{aligned}$$

(A.95)

where now $f_{\pm ,\lambda }$ stands for the newform with W-eigenvalue $\pm 1$ and $T_2$-eigenvalue $\lambda$ and $F_{\pm ,\lambda }$ stands for an associated meromorphic partner. The latter are chosen such that the leading coefficient in the Fourier expansion is 1 and the coefficient of q vanishes. The maximal denominators in these expansions can be read off from the integer by which we divide $\delta _\pm$. Explicitly the action of the Hecke operator $T_2$ on the meromorphic representatives is given by

$$\begin{aligned} F_{+,1}|_4 (T_2 - 1)&= D^3 \ \frac{e_+}{h_+} \left( -\frac{1}{27}\phi _+^2+\frac{1}{9} \right) \nonumber \\ F_{+,4}|_4 (T_2-4)&= D^3 \ \frac{e_+}{h_+} \left( -\frac{1}{27}\phi _+^2+\frac{11}{54} \right) \nonumber \\ F_{+,-4}|_4 (T_2+4)&= D^3 \ \frac{e_+}{h_+} \left( -\frac{1}{27}\phi _+^2+\frac{7}{27} \right) \nonumber \\ F_{-,-1}|_{4} (T_2+1)&= D^3 \ \frac{e_+}{h_-} \left( -\frac{1}{8}\phi _++\frac{1}{4} \right) \nonumber \\ F_{-,-4}|_{4} (T_2+4)&= D^3 \ \frac{e_+}{h_-} \left( -\frac{1}{8}\phi _++\frac{1}{4} \right) . \end{aligned}$$

(A.96)

1.3 Zeta functions and the motivic point of view

There are different cohomology groups one can associate with smooth projective varieties defined over $\mathbb {Q}$ (or more generally any number field). These can be used to define periods and zeta functions, the latter being related to the number of points over finite fields. In the following we briefly discuss these objects and sketch the idea of motives, which capture the cohomological structure of varieties. As the most important example for this paper, we explain that there are motives attached to Hecke eigenforms.

1.3.1 Hodge theory and periods

Let X be a smooth projective variety of dimension d defined over $\mathbb {Q}$. Viewing X as a complex manifold, we have for each integer r between 0 and 2d the rth homology group $H_r(X(\mathbb {C}),\mathbb {Z})$ whose elements are represented by closed r-dimensional chains modulo boundaries of $(r+1)$-dimensional chains. The dimension of this space is the rth Betti number $b_r(X)$. Considering the cochain complex we also get the associated cohomology groups $H^r(X(\mathbb {C}),\mathbb {Z})$. By de Rham’s theorem, we can represent elements of $H^r(X(\mathbb {C}),\mathbb {Z})$ by elements of the de Rham cohomology group $H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C})$ whose elements are represented by closed r-forms modulo exact r-forms. More concretely, by Stokes’s theorem, the integration of differential forms over chains gives a well defined pairing

$$\begin{aligned} \int : \ \ H_r(X(\mathbb {C}),\mathbb {Z}) \otimes _{\mathbb {Z}} H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C}) \rightarrow \mathbb {C} \end{aligned}$$

(A.97)

and by de Rham’s theorem this pairing is non-degenerate. This induces an isomorphism

$$\begin{aligned} H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C}) \; \xrightarrow {\sim } \; H^r(X(\mathbb {C}),\mathbb {Z}) \otimes _{\mathbb {Z}} \mathbb {C} . \end{aligned}$$

(A.98)

The complex structure of $X(\mathbb {C})$ further allows us, by a theorem of Hodge, to decompose $H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C})$ into subspaces whose elements can be represented by forms of Hodge type (p, q) with $p+q=r$. This gives the Hodge decomposition

$$\begin{aligned} H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C}) = \sum _{p+q=r} H^{p,q}(X(\mathbb {C})) . \end{aligned}$$

(A.99)

Up to now nothing required X to be defined over $\mathbb {Q}$. This changes now as we want to to use the pairing of homology and cohomology to define periods. To do this, we replace the complex vector spaces $H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C})$ by the algebraic de Rham cohomology groups $H^r_{\text {dR}}(X)$ which are vector spaces over $\mathbb {Q}$. These were defined by Grothendieck [96] as the hypercohomology groups of a certain algebraic de Rham complex. In particular, Grothendieck proves that there is a natural isomorphism

$$\begin{aligned} H^r_{\text {dR}}(X) \otimes _{\mathbb {Q}}\mathbb {C} \; \cong \; H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C}) , \end{aligned}$$

(A.100)

called the comparison isomorphism. For the algebraic de Rham cohomology groups, we do not have a Hodge decomposition but only a Hodge filtration

$$\begin{aligned} F^rH^r_{\textrm{dR}}(X) \; \subseteq \; F^{r-1}H^r_{\textrm{dR}}(X) \; \subseteq \; \cdots \; \subseteq \; F^0H^r_{\textrm{dR}}(X) = H^r_{\textrm{dR}}(X) \end{aligned}$$

(A.101)

which, with respect to the comparison isomorphism, is compatible with the Hodge filtration of $H_{\text {dR}}^r(X(\mathbb {C}),\mathbb {C})$ induced by the Hodge decomposition, i.e.

$$\begin{aligned} F^kH^r_{\textrm{dR}}(X) \otimes _{\mathbb {Q}} \mathbb {C} \; \cong \; \bigoplus _{p \ge k} H^{p,r-p}(X(\mathbb {C})) . \end{aligned}$$

(A.102)

For example, if X is an elliptic curve defined over $\mathbb {Q}$, a basis of $H^1_{\textrm{dR}}(X)$ is given by a differential $\omega$ of the first kind and a differential $\eta$ of the second kind, both defined over $\mathbb {Q}$. While $\omega$ has Hodge type (1, 0), $\eta$ will be a mix of the Hodge types (1, 0) and (0, 1) and is canonically defined only up to multiplication by a non-zero rational number and addition of a rational multiple of $\omega$.

Using the comparison isomorphism, we can now define the non-degenerate pairing

$$\begin{aligned} \int : \ \ H_r(X(\mathbb {C}),\mathbb {Z}) \otimes _{\mathbb {Z}}H^r_{\text {dR}}(X) \; \rightarrow \; \mathbb {C} . \end{aligned}$$

(A.103)

By choosing a basis for $H_r(X(\mathbb {C}),\mathbb {Z})$ and $H^r_{\text {dR}}(X)$ this gives rise to a complex $b_r(X) \times b_r(X)$ matrix called the period matrix. The period matrix is unique up to multiplication by a unimodular integer matrix from the left and multiplication by an invertible rational matrix from the right. The Hodge filtration of the algebraic de Rham cohomology groups further induces a filtration of the periods which allows to restrict possible matrices multiplied from the right to lower triangular matrices.

1.3.2 Reduction modulo primes and zeta functions

Let X as before be a smooth projective variety of dimension d defined over $\mathbb {Q}$. Since X is given as a subspace of some projective space by equations with rational coefficients, we can reduce these defining equations (after multiplication by an integer to clear the denominators) modulo any prime p, leading to a variety $X_p:= X/\mathbb {F}_p$ defined over $\mathbb {F}_p$. We restrict to the case that this variety is smooth, which happens for all but finitely many p, called the good primes. The remaining primes are called bad primes. For any $n \ge 1$ we consider the number $\# X_p(\mathbb {F}_{p^n})$ of solutions of the defining equations with the variables taking their values in the field $\mathbb {F}_{p^n}$. The local zeta function of $X_p$ is a generating function of these numbers

$$\begin{aligned} Z(X_p,T) = \exp \left( \sum _{n=1}^\infty \# X_p(\mathbb {F}_{p^n}) \frac{T^n}{n} \right) . \end{aligned}$$

(A.104)

A deep theorem says that $Z(X_p,T)$ is not just a power series but a rational function in T with integral coefficients. Moreover, Weil conjectured that this rational function has the form

$$\begin{aligned} Z(X_p,T) = \prod _{r=0}^{2d} P_r(X_p,T)^{(-1)^{r+1}} \end{aligned}$$

(A.105)

where $P_r(X_p,T)$ is a polynomial of degree $b_r(X)$ with integral coefficients and with all roots of absolute value $p^{-r/2}$ (“local Riemann hypothesis”) and satisfies the functional equation

$$\begin{aligned} P_{2d-r}( X_p,1/p^d T ) = \pm P_r(X_p,T)/(p^{d/2}T)^{b_r(X)} . \end{aligned}$$

(A.106)

He further conjectured that it should be possible to prove this by finding an appropriate cohomology theory for the variety $X_p$ defined over $\mathbb {F}_p$. This was later realized through the work of Grothendieck, Artin and others by introducing, in general for any smooth projective variety V defined over any field K, the $\ell$-adic cohomology group $H^r(\overline{V},\mathbb {Q}_\ell )$ for any prime $\ell \ne \text {char } K$. Here, $\overline{V}$ stands for the variety V regarded as a variety over the algebraic closure $\overline{K}$. In particular, the Galois group $\text {Gal}(\overline{K}/K)$ naturally acts on $\overline{V}$ and this action induces an action on $H^r(\overline{V},\mathbb {Q}_\ell )$. In the case $V=X_p$ and $K=\mathbb {F}_p$, the Galois group is topologically generated by the Frobenius automorphism $\text {Fr}_p: x \mapsto x^p$ and the fixed points of the nth power of $\text {Fr}_p$ on $X_p(\overline{\mathbb {F}_p})$ are precisely the points defined over $\mathbb {F}_{p^n}$. This can be used to relate $\# X_p(\mathbb {F}_{p^n})$ to the traces of the Frobenius automorphism, since, as proven by Grothendieck, the Lefschetz trace formula can be applied also to the $\ell$-adic cohomology groups, and one obtains

$$\begin{aligned} \# X_p(\mathbb {F}_{p^n}) = \sum _{r=0}^{2d} (-1)^r \text {tr}((\text {Fr}_p^*)^n \, | \, H^r(\overline{X_p},\mathbb {Q}_\ell ) ) . \end{aligned}$$

(A.107)

A direct consequence is that the local zeta function has the form

$$\begin{aligned} Z(X_p,T) = \prod _{r=0}^{2d} \det (1-T\text {Fr}_p^*\, | \, H^r(\overline{X_p},\mathbb {Q}_\ell ))^{(-1)^{r+1}} . \end{aligned}$$

(A.108)

In particular, the product on the right is independent of the chosen prime $\ell$. Because of the local Riemann hypothesis, which was proven by Deligne, the same holds for each factor, giving the desired polynomial $P_r(X_p,T) \in \mathbb {Z}[T]$.

The considerations above apply to any smooth projective variety defined over $\mathbb {F}_p$ and not only to the reduction $X_p$ of a variety X defined over $\mathbb {Q}$. However, using that we have a global variety X defined over $\mathbb {Q}$ allows us to define the $\ell$-adic cohomology group $H^r(\overline{X},\mathbb {Q}_l)$ for all primes l. An important fact is that there is a natural comparison isomorphism

$$\begin{aligned} H^r(\overline{X},\mathbb {Q}_l) \; \cong \; H^r(X(\mathbb {C}),\mathbb {Z}) \otimes _{\mathbb {Z}} \mathbb {Q}_\ell . \end{aligned}$$

(A.109)

In particular, this implies that $P_r(X_p,T)$ is a polynomial of degree $b_r(X)$. Another important theorem is that for all good primes $p \ne l$ there is a natural isomorphism

$$\begin{aligned} H^r(\overline{X},\mathbb {Q}_l) \; \cong \; H^r(\overline{X_p},\mathbb {Q}_l) . \end{aligned}$$

(A.110)

The Frobenius automorphism $\text {Fr}_p$ then corresponds to a well-defined conjugacy class in the action of $\text {Gal}(\overline{\mathbb {Q}}/\mathbb {Q})$ on $H^r(\overline{X},\mathbb {Q}_l)$, which we also denote by $\text {Fr}_p$, and we have

$$\begin{aligned} P_r(X_p,T) = \det (1-T\text {Fr}_p^*\, | \, H^r(\overline{X_p},\mathbb {Q}_\ell )) = \det (1-T\text {Fr}_p^*\, | \, H^r(\overline{X},\mathbb {Q}_\ell )) . \end{aligned}$$

(A.111)

If p is a bad prime one can still associate a conjugacy class to $\text {Fr}_p$ but this is only well defined up to elements in an inertia subgroup $I_p$. For these primes one defines

$$\begin{aligned} P_r(X_p,T) = \det (1-T\text {Fr}_p^*\, | \, H^r(\overline{X},\mathbb {Q}_\ell )^{I_p}) , \end{aligned}$$

(A.112)

whose degree in T is at most $b_r(X)$.

The fact that all local zeta functions come from the same variety X allows us to define the Hasse–Weil zeta function

$$\begin{aligned} \zeta (X/\mathbb {Q},s) = \prod _p Z(X/\mathbb {F}_p,p^{-s}) \qquad (\text {Re }s \gg 0) \end{aligned}$$

(A.113)

which may also be written as an alternating product of the L-functions

$$\begin{aligned} L_r(X/\mathbb {Q},s) = \prod _p P_r(X/\mathbb {F}_p,p^{-s})^{-1} \qquad (\text {Re }s \gg 0) . \end{aligned}$$

(A.114)

One of the most important conjectures in modern arithmetic algebraic geometry is that each $L_r$ has remarkable analytic properties. For example, it is expected that $L_r$ can be analytically continued to a meromorphic function on the complex plane which has a functional equation with respect to the symmetry $s \mapsto r+1-s$. For a few varieties, these properties can be proven but in almost all cases they are conjectural. For a more detailed treatment, we refer to [97].

We finish with some remarks regarding the computation of the local zeta function. We have seen that the local zeta function can be obtained by either counting the number of points over finite fields or studying $\ell$-adic cohomology groups. In practice these methods quickly become infeasible for complicated varieties and large primes p. However, there are also p-adic cohomology theories which allow a more efficient computation. A good review explaining how these can be used to compute the local zeta function is [98]. Given a family of varieties one may further use the periods to compute the local zeta function very efficiently. This was first considered by Dwork and for one-parameter Calabi–Yau threefolds this is explained for example in [15].

1.3.3 The motivic point of view

The idea of motives was proposed by Grothendieck to capture the cohomological structure of varieties. We want to briefly explain this idea without going much into detail. For more details we refer to [99]. We start by explaining geometric motives. Let X be a smooth projective variety of some dimension n. For simplicity, we assume that X is defined over $\mathbb {Q}$ (more generally one could consider any number field). In A.3.1 we recalled that for every integer $0 \le r \le 2n$ we can associated different cohomology groups with X:

• by considering the complex points on X we obtain a topological space $X(\mathbb {C})$ which gives rise to the Betti cohomology group $H^r(X(\mathbb {C}),\mathbb {Z})$,

• using the structure of X as a variety defined over $\mathbb {Q}$ we obtain the algebraic de Rham cohomology group $H^r_{\text {dR}}(X)$ with the usual Hodge filtration,

• letting $\overline{X}$ be the variety X regarded as a variety over $\overline{\mathbb {Q}}$ one obtains for any prime $\ell$ the $\ell$-adic cohomology group $H^r(\overline{X},\mathbb {Q}_\ell )$ upon which $\text {Gal}(\overline{\mathbb {Q}}/\mathbb {Q})$ acts.

We also saw that these are not unrelated, e.g. there are comparison isomorphisms between $H^r(X(\mathbb {C}),\mathbb {Z}) \otimes _{\mathbb {Z}} \mathbb {C}$ and $H^r_{\text {dR}}(X) \otimes _{\mathbb {Q}} \mathbb {C}$ and between $H^r(X(\mathbb {C}),\mathbb {Q}) \otimes _{\mathbb {Q}} \mathbb {Q}_\ell$ and $H^r(\overline{X},\mathbb {Q}_\ell )$.

The simplest example of a geometric motive is the vector space $V = H^r(X(\mathbb {C}),\mathbb {Q})$ together with the Hodge decomposition on $V \otimes _{\mathbb {Q}} \mathbb {C}$ and the action of $\text {Gal}(\mathbb {\overline{Q}}/\mathbb {Q})$ on $V \otimes _{\mathbb {Q}}\mathbb {Q}_\ell$ for primes $\ell$. More generally, consider an algebraic cycle $\gamma \in Z^n(X \times X)$ defined over $\mathbb {Q}$ (an example of a correspondence). This induces an element in $H^{2n}(X \times X)$ and using the Künneth isomorphism and Poincaré duality this gives elements $\sigma _r \in \text {End}(H^r(X(\mathbb {C}),\mathbb {Q}),H^r(X(\mathbb {C}),\mathbb {Q}))$ for any $0 \le r \le 2n$. The same can be done for the algebraic de Rham cohomology groups and the $\ell$-adic cohomology groups. If some $\sigma _r$ is a projector we now say that the kernel (and hence also the image) of $\sigma _r$ is a geometric motive. This subspace is automatically compatible with the Hodge decomposition and the action of $\text {Gal}(\mathbb {\overline{Q}}/\mathbb {Q})$. The weight of such a motive is defined to be r and can be read off from the motive itself by the fact that the eigenvalues of $\text {Fr}_p^*$ have absolute value $p^{r/2}$.

Conjecturally, any linear subspace $V\subseteq H^r(X(\mathbb {C}),\mathbb {Q})$ that is compatible with the Hodge decomposition and the action of $\text {Gal}(\mathbb {\overline{Q}}/\mathbb {Q})$ defines a geometric motive, i.e. is cut out by some correspondence. Even stronger, Hodge-like conjectures and Tate-like conjectures would imply that a linear subspace $V \subseteq H^r(X(\mathbb {C}),\mathbb {Q})$ is already a geometric motive if it is compatible with the Hodge decomposition or with the Galois action. This can be summarized in the following diagram:

More generally, a motive can be thought of as a suitable collection of vector spaces (equipped with a Hodge decomposition and an action of $\text {Gal}(\overline{\mathbb {Q}}/\mathbb {Q})$ with additional compatibilities). It should always be representable as a geometric motive contained in a cohomology group of some variety, but the choice of the geometric realization is not part of the definition of the motive. Examples of motives that do not refer to specific varieties are hypergeometric motives, for which we refer to the survey article by Roberts and Villegas [100], and motives associated with modular forms, which we now describe.

1.3.4 The motives attached to Hecke eigenforms

In this final subsection we explain that there are motives attached to arbitrary newforms, the weight of the motive being one less than that of the modular form. The point we want to stress is that the motive $V_f$ attached to a newform f is an intrinsically defined object, independent of any specific geometric realization: it always has a geometric realization, as a consequence of the Eichler-Shimura theory if $k=2$ and of the work of Deligne if $k>2$, as explained below, but in general it can have others. The situation of relevance to this paper is that of the motives attached to newforms of weight 4 and 2 occurring in the 3rd cohomology group of some Calabi–Yau threefolds (fibers over conifold points and attractor points of hypergeometric families), but there are many other examples in the literature. For instance, it was shown by Ron Livné in the 1980 s that the $L_7$-factor of the Hasse–Weil zeta function of the 7-dimensional variety $\{(x_1:\cdots :x_{10})\in \mathbb {P}^9 \mid \sum _i x_i = \sum _i x_i^3=0\}$ splits as a product of a number of Riemann zeta functions and the L-function of the unique newform of level 10 and weight 4. For more discussion and other examples we refer to [101] (pp. 150–151), [102], and [103].

Geometric realization of $\varvec{V_f}$ for all newforms The simplest situation arises for modular forms of weight 2 and some level N. In this case one can consider the modular curve ${X_0(N) = \Gamma _0(N) \backslash \overline{\mathfrak {H}}}$ and there is a canonical isomorphism

$$\begin{aligned} \mathbb {S}_2(\Gamma _0(N)) \;&\xrightarrow {\sim } \; H^1_{\text {dR}}(X_0(N),\mathbb {C}) \nonumber \\ [F] \;&\mapsto \; [2\pi i F \, \hbox {d}\tau ] . \end{aligned}$$

(A.115)

In fact, $X_0(N)$ can be given the structure of a smooth projective variety defined over $\mathbb {Q}$ and if one restricts to classes that can be represented by forms in $S_2^{[\infty ]}(\Gamma _0(N))$ with rational Fourier coefficients this gives an isomorphism with $H^1_{\text {dR}}(X_0(N))$. Hence we have a natural motive $V = H^1(X_0(N),\mathbb {Q})$ we can consider. For any divisor $N'$ of N there are $[\Gamma _0(N'):\Gamma _0(N)]$ correspondences on $X_0(N) \times X_0(N)$ which give a splitting $V = V^{\text {new}} \oplus V^{\text {old}}$ corresponding to the splitting into old forms and new forms. There are Hecke correspondences which further split $V^{\text {new}}$ so that attached to any newform f with rational Hecke eigenvalues (the case of more general coefficients is similar) we obtain a 2-dimensional geometric motive $V_f$. From the work of Eichler and Shimura it follows that for all primes $p \not \mid N$ and $\ell \ne p$

$$\begin{aligned} \det (1-\text {Fr}_p^* T | V_f \otimes _{\mathbb {Q}} \mathbb {Q}_\ell ) = 1-a_pT+pT^2 \end{aligned}$$

where $a_p$ is the eigenvalue of f under $T_p$. We conclude that attached to f there is a geometric motive $V_f$ so that the periods of $V_f$ are the periods of quasiperiods and f and the traces of the Frobenius operators are just the eigenvalues of the Hecke operators.

For newforms $f\in S_k(\Gamma _0(N))$ of weight $k>2$, Deligne [11] showed that the Hecke eigenvalues coincide with the eigenvalues of the Frobenius operators in the $(k-1)$-st cohomology group of an appropriate Kuga-Sato variety, defined as a suitable compactification of a fiber bundle over $\Gamma _0(N) \backslash \mathfrak {H}$ whose fiber over a point $\tau$ is the $(k-2)$-nd Cartesian product of the level N elliptic curve $E_\tau$. Scholl [12] used this construction to associate a motive with $V_f$ with f. If f has rational Hecke eigenvalues the attached motive $V_f$ is again 2-dimensional and the periods of $V_f$ are given by the periods and quasiperiods of f.

We remark that newforms of weight 1 (defined either for suitable subgroups of $\Gamma _0(N)$ or with a character in the slash operator) are also motivic. Geometrically these motives are not very interesting since the relevant varieties are 0-dimensional. As an example we consider the newform of level 23 defined by $f(\tau ) = \eta (\tau )\eta (23\tau )$. This is associated with the variety defined by $x^3-x-1$ and this manifests in the number of roots of this polynomial over the finite field $\mathbb {F}_p$ for primes $p\ne 23$ being $a_p+1$ where $a_p$ is the eigenvalue of the Hecke operator $T_p$. This example was given by Blij in [104].

Correspondences between different geometric realizations Conjecturally, two different geometric realizations of motives must be related by a correspondence. We give one example in Sect. 4 where we construct a correspondence between a conifold fiber of a hypergeometric family of Calabi–Yau threefolds and a Kuga-Sato variety associated with the unique newform $f \in S_4(\Gamma _0(8))$. The Tate conjecture would imply that there must be a correspondence already if two Galois representations coincide. While the construction of correspondences can be difficult, theorems of Faltings and Serre allow to establish the equality of two Galois representations by comparing finitely many Frobenius traces. E.g. for the conifold fiber of the quintic this was used by Schoen [9] to prove the equality of the associated Galois representation with that of the relevant newform of level 25 and weight 4.

Appendix: Computational Results

In the main part of this paper we considered 16 newforms of weight 4 (associated with 14 conifold points and 2 attractor points) and 2 newforms of weight 2 (associated with 2 attractor points). The Atkin-Lehner eigenvalues and beginning of the q-expansions of these forms can be found in Table 8 (for the modular forms associated with conifold points) and Table 9 (for the modular forms associated with attractor points). In the following we explain how we computed the periods and quasiperiods associated with these forms.

Table 8 Atkin–Lehner eigenvalues and the q-expansions of newforms of weight 4 associated with conifolds

Table 9 Atkin–Lehner eigenvalues and the q-expansions of newforms associated with attractors

Table 10 The q-expansions of meromorphic partners of weight 4 associated with conifolds

Table 11 The q-expansions of meromorphic partners associated with attractors

Table 12 Period polynomials and approximate values of periods and quasiperiods for newforms of weight 4 associated with conifolds and for chosen $\gamma \in \Gamma _0(N)$

Table 13 Period polynomials and approximate values of periods and quasiperiods for newforms associated with attractors and for chosen $\gamma \in \Gamma _0(N)$

Table 14 For each given level N the unique normalized form $h\in M_{k_h}(\Gamma _0^*(N))$ such that $k_h$ is as small as possible and h has maximal vanishing order at $\infty$

For each newform f of level N and weight k we choose the Eichler integral $\widetilde{f} = {\widetilde{f}}_\infty$ as defined in (A.15) and then compute the period polynomials $r_f(\gamma )$ for a set of generators of $\Gamma _0(N)$ by using (A.16) with $\tau _0$ chosen so that the imaginary parts of $\tau _0$ and $\gamma ^{-1}\tau _0$ are as large as possible. These period polynomials can then be written as

$$\begin{aligned} r_f(\gamma ) = \omega _f^+ \hat{r}_f^+(\gamma )+\omega _f^- \hat{r}_f^-(\gamma ) \end{aligned}$$

(B.1)

with $\hat{r}_f^\pm \in Z^1(\Gamma _0(N),V_{k-2}(\mathbb {Q}))^\pm$ and we choose the periods $\omega _f^\pm$ so that all $\hat{r}_f^\pm$ have integral coefficients and do not have any non-trivial common divisor. This makes the periods unique up to a sign which we fix by requiring $\omega _f^+, \text {Im} \ \omega _f^->0$. We list numerical values for the periods and $\hat{r}_f^\pm (\gamma )$ for a chosen $\gamma \in \Gamma _0(N)$ in Table 12 and Table 13.

To compute the quasiperiods associated to a normalized Hecke eigenform f of level N and weight k we first find a meromorphic form F such that [F] has the same Hecke eigenvalues as f. We make the ansatz

$$\begin{aligned} F=\frac{g}{h} \end{aligned}$$

(B.2)

where $g\in S_{k+k_h}(\Gamma _0(N))$ has the same Atkin-Lehner eigenvalues as f and $h\in M_{k_h}(\Gamma _0^*(N))$ is chosen such that $k_h$ is as small as possible and h has the maximal vanishing order at $\infty$. Such a form necessarily has to be an eta quotient and the forms are explicitly given in Table 14. We then determine g so that [F] has the same Hecke eigenvalues as f and normalize F so that the quasiperiods fulfill

$$\begin{aligned} \omega _f^+ \eta _F^--\omega _f^-\eta _F^+ = (2\pi i)^{k-1} . \end{aligned}$$

(B.3)

This makes [F] unique up to the addition of rational multiples of [f]. For the modular forms associated with the conifold points we fix this by requiring that the ratios $\eta _f^\pm /e^\pm$ are rational and for modular forms of weight 2 (or 4) associated with the attractor points we fix this by requiring that the projection of $\Pi ''(z_*)$ (or $\Pi '''(z_*)$) on the Hodge structure (1, 2) (or (0, 3)) are given by rational linear combinations of the quasiperiods (or rational linear combinations multiplied by $2\pi i$). The beginning of the q-expansions of our choice of meromorphic forms can be found in Table 10 and Table 11. The resulting numerical values for the quasiperiods are given in Table 12 and Table 13. We provide a supplementary Pari file containing more detailed data at [105].