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On quaternionic ordinary families of modular forms and $p$-adic $L$-functions

Matteo Longo, Paola Magrone, Eduardo Rocha Walchek

Abstract

We use Serre--Tate expansions of modular forms to construct power series attached to quaternionic ordinary families of modular forms. We associate to these power series a big $p$-adic $L$-function interpolating the $p$-adic $L$-functions constructed by Burungale and Magrone at classical specializations. A crucial ingredient is the generalization of some results of Ohta to the quaternionic setting.

On quaternionic ordinary families of modular forms and $p$-adic $L$-functions

Abstract

We use Serre--Tate expansions of modular forms to construct power series attached to quaternionic ordinary families of modular forms. We associate to these power series a big -adic -function interpolating the -adic -functions constructed by Burungale and Magrone at classical specializations. A crucial ingredient is the generalization of some results of Ohta to the quaternionic setting.

Paper Structure

This paper contains 35 sections, 30 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Shimura curves
  3. Quaternion algebras
  4. QM abelian surfaces
  5. Shimura curves with naı̈ve level structures
  6. Shimura curves with Drinfeld level structures
  7. Igusa curves
  8. Special fibers
  9. Modular forms
  10. Modular forms on Shimura curves
  11. Hecke operators
  12. Atkin--Lehner involution
  13. $p$-stabilizations and $p$-depletions
  14. Katz modular forms
  15. Power series expansions
  16. ...and 20 more sections

Key Result

Theorem 2.7

Let $m\geq 0$ be an integer. The functor which takes a $\mathbb{Z}[1/dD]$-scheme $S$ to the set of isomorphism classes of such triples $(A,\iota,\alpha)$ consisting of a QM abelian surface $(A,\iota)$ equipped with a naïve level $V_1(d)$ structure over $S$ is representable by a $\mathbb{Z}[1/dD]$-sc

Theorems & Definitions (84)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof : References for the proof
  • Definition 2.8
  • Theorem 2.9
  • ...and 74 more