Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A $p$-adic cohomological approach to congruences of meromorphic modular forms

Paolo Bordignon

TL;DR

This work develops a $p$-adic cohomological framework to explain congruences between Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues from poles on modular and Shimura curves. By interfacing rigid cohomology, log-crystalline theory, and the Gysin sequence, it relates Frobenius actions on de Rham cohomology at prescribed poles to the $U_p$-action on overconvergent meromorphic modular forms, uniformly across curves with smooth integral models over $\mathbb{Z}_p$. The construction yields a Frobenius-stable lattice and an explicit Frobenius lift, enabling deductions of $q$-expansion congruences and connecting concrete examples (e.g., Zhang’s congruences) to the cohomological eigenspaces of symmetric powers of pole-associated abelian varieties. The framework extends to Shimura curves and clarifies how $p$-adic Frobenius data governs modular-form congruences, providing a robust method to derive arithmetic congruences from geometric cohomology.

Abstract

We study congruences relating Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues of elliptic curves corresponding to their poles. We develop a $p$-adic cohomological framework that interprets these congruences via the interaction between the rigid cohomology of modular curves and the crystalline structure of the associated elliptic curves. Using comparison theorems and the Gysin sequence, we relate the Frobenius actions in cohomology to the $U_p$-operator acting on spaces of overconvergent modular forms. Our approach applies uniformly to both modular curves and Shimura curves admitting smooth integral models over $\mathbb{Z}_p$.

A $p$-adic cohomological approach to congruences of meromorphic modular forms

TL;DR

This work develops a -adic cohomological framework to explain congruences between Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues from poles on modular and Shimura curves. By interfacing rigid cohomology, log-crystalline theory, and the Gysin sequence, it relates Frobenius actions on de Rham cohomology at prescribed poles to the -action on overconvergent meromorphic modular forms, uniformly across curves with smooth integral models over . The construction yields a Frobenius-stable lattice and an explicit Frobenius lift, enabling deductions of -expansion congruences and connecting concrete examples (e.g., Zhang’s congruences) to the cohomological eigenspaces of symmetric powers of pole-associated abelian varieties. The framework extends to Shimura curves and clarifies how -adic Frobenius data governs modular-form congruences, providing a robust method to derive arithmetic congruences from geometric cohomology.

Abstract

We study congruences relating Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues of elliptic curves corresponding to their poles. We develop a -adic cohomological framework that interprets these congruences via the interaction between the rigid cohomology of modular curves and the crystalline structure of the associated elliptic curves. Using comparison theorems and the Gysin sequence, we relate the Frobenius actions in cohomology to the -operator acting on spaces of overconvergent modular forms. Our approach applies uniformly to both modular curves and Shimura curves admitting smooth integral models over .
Paper Structure (10 sections, 7 theorems, 76 equations)

This paper contains 10 sections, 7 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. An important note on a recent result
  3. Acknowledgments
  4. Geometric setting
  5. Rigid cohomological description
  6. Integral Frobenius structure
  7. Explicit Frobenius lift
  8. Arithmetic applications
  9. Congruences on $q$-expansion
  10. An explicit example - Zhang's congruences

Key Result

Theorem \ref{Theorem characteristic polynomial of Up action}

Let $X$ be a modular curve (elliptic or quaternionic) of level $N$ defined over $\mathbb{Z}_p$, with $N>3$, $p>3$ and $(N,p)=1$. Fix a $\mathbb{Z}_p$-point $\alpha$ on the modular curve. Let $k$ be a positive integer with $p>k+1$, let $P(X),Q(X)\in \mathbb{Q}_p[X]$ be respectively the characteristic with $d=\dim_{\mathbb{Q}_p}H^1_{\mathrm{dR}}(X^{\mathrm{rig}},\mathcal{H}_k)=\dim_{\mathbb{Q}_p}M_{

Theorems & Definitions (14)

  • Theorem \ref{Theorem characteristic polynomial of Up action}
  • Lemma 1
  • proof
  • Remark 1: Parallel transport of eigenbasis
  • Theorem 1: coleman1996classical Theorem 5.4
  • Lemma 2
  • Theorem A
  • proof
  • Remark 2: Hecke operators
  • Remark 3: Slope of $p$-newforms
  • ...and 4 more