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%P
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Book
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Chapter
Special Topics
Let ω 1 and ω 2 be two non-zero complex numbers which are linearly independent over ℝ.
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Chapter
The Modular Group
Let R be a commutative ring with identity. Show that the set $$ {\text{SL}}_{2} (R) = \left\{ {\left( {\begin...
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Chapter
Modular Forms of Level One
Show that the punctured fundamental neighborhood $$ U_{c} = \{ z \in {\mathbb{H}}:\text{Im} (z) > c\} $$ ...
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Chapter
Jacobi’s q-series
It is not clear exactly how to define a q-series. Some experts humorously suggest that it is any power series in q. To some extent this may be true. However, one can say that part of the theory is connected with ...
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Chapter
Modular Forms of Higher Level
Let Γ be a congruence subgroup of \( {\text{SL}}_{2} ({\mathbb{Z}}) \) of level N. Recall that ...
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Chapter
Special Topics
This is a consequence of Liouville’s theorem which says that any bounded entire function is constant. Since f is holomorphic and completely determined by its values on the compact region
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Chapter
The Upper Half-Plane
Let ℍ denote the upper half-plane, $$ {\mathbb{H}} = \{ z \in {\mathbb{C}}:\text{Im} (z) > 0\} , $$ viewed ...
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Chapter
Hecke Operators of Higher Level
Let Γ1 and Γ2 be subgroups of \( {\text{GL}}_{2}^{ + } \left( {\mathbb{Q}} \right) \) and let ...
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Chapter
The Ramanujan τ-function
In his fundamental paper of 1916, Ramanujan introduced the τ-function as being the coefficients in the power series expansion of the infinite product
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Chapter
The Petersson Inner Product
Let Γ be a congruence subgroup of \( {\text{SL}}_{2} \left( {\mathbb{Z}} \right) \) and ...
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Chapter
Dirichlet Series and Modular Forms
The general Dirichlet series is a discrete analog of the more familiar Laplace transform of a function in the theory of complex variables.
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Chapter
Jacobi’s q-series
Show that $$ \varepsilon_{q} (x) - \varepsilon_{q} \left( {\frac{x}{q}} \right) = \frac{x...
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Chapter
The Upper Half-Plane
Show that $$ \left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right)z = \frac{az + b}{cz + d} $$ ...
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Chapter
The Ramanujan τ-Function
In his fundamental paper of 1916, Ramanujan introduced the τ-function as being the coefficients in the power series expansion of the infinite product.
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Chapter
The Modular Group
The (full) modular group SL2(ℤ) plays a pivotal role in the theory of modular forms. One also considers PSL2(ℤ) = SL2(ℤ)/{±I}.
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Chapter
The Petersson Inner Product
Let Γ be a congruence subgroup of \( {\text{SL}}_{2} \left( {\mathbb{Z}} \right) \) and ...
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Chapter
Modular Forms of Level One
We will use the topology on the extended upper half-plane ℍ* to define what it means for f(z) to be holomorphic at the cusp i∞.
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Chapter
Dirichlet Series and Modular Forms
Writing s = σ + it, with σ > 1, and \( t \in {\mathbb{R}} \) , we have
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Chapter
Modular Forms of Higher Level
Let Γ be a congruence subgroup of \( {\text{SL}}_{2} ({\mathbb{Z}}) \) of level N. Recall that ...
