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The Eigencurve of Modular Symbols

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The Eigenbook

Part of the book series: Pathways in Mathematics ((PATHMATH))

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Abstract

Stevens’s overconvergent modular symbols can replace Coleman’s overconvergent modular forms in the construction of the eigencurve. We give in Sect. 7.1 a detailed version of this construction. We then (Sect. 7.2) use Chenevier’s comparison theorem (see Sect. 3.8) to prove that this eigencurve is essentially the same as the Coleman-Mazur eigencurve. This construction provides over the eigencurve a natural module of distribution-valued modular symbols which is closely related to L-functions. In the next Sect. 7.3, we offer a detailed discussion of the points of the eigencurve that correspond to classical modular forms. Those points, and their neighborhoods in the eigencurve, are the main objects of interest for arithmetical applications. Section 7.4 introduces and studies one of the last remaining important characters of our intrigue, the family of Galois representations that the eigencurve carry. Using this family and Galois cohomology arguments, we study the local geometry of the eigencurve at classical points, focussing on questions of smoothness and étaleness over the weight space. Finally, we briefly discuss the most important results and questions concerning the global geometry of the eigencurve.

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Notes

  1. 1.

    For the definition of linked, see Sect. 3.5.

  2. 2.

    For the definition of a generalized elliptic curve, see [53, Chapter II]. Let us just recall that the smooth locus E reg of a generalized elliptic curve is a group scheme over S, that E is smooth if and only if it is an elliptic curve over S, and that non-smooth generalized elliptic curves over fields correspond to cusps in the moduli schemes of generalized elliptic curves.

  3. 3.

    The definition of such a structure can be found in [77] but ww will not use it here.

  4. 4.

    For the definition of the generalized elliptic curve Tate(q N), see [53].

  5. 5.

    To understand this definition, recall that E p−1 lifts the Hasse-invariant modulo p. Thus \(\tilde X(\Gamma ,0)\) consist of all points (E, H, α) such that |E p−1(E, η)| = 1, that is such that either E is a non-smooth generalized elliptic curve (i.e. (E, α, H) is a cusp of ), or E is a true elliptic curve and the reduction of E mod p is ordinary. If v > 0, then \(\tilde X(\Gamma ,v)\) is a slightly larger affinoid containing also points (E, H, α) where E has supersingular reduction.

  6. 6.

    French très classique. This terminology seems to be due to Chenevier.

  7. 7.

    This was at first a question of Greenberg, now become a folklore conjecture. See [61].

  8. 8.

    A form f of level prime to p is regular at p if its two p-adic refinements f α and f β are distinct.

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  1. Department of Mathematics, Brandeis University, Waltham, MA, USA

    Joël Bellaïche

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Bellaïche, J. (2021). The Eigencurve of Modular Symbols. In: The Eigenbook. Pathways in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-77263-5_7

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