Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms

Sipra Maity, Rupam Barman

Abstract

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we construct two explicit families of eta-quotients, which we call the $\mathbb{K}_4$ and $\mathbb{K}_5$ functions, from the hypergeometric background. These $\mathbb{K}_4$ and $\mathbb{K}_5$ functions are constructed using the theory of weight $1/2$ Jacobi theta functions and their cubic analogues, respectively. Using these constructions, we then express the Fourier coefficients of certain Hecke eigenforms of weight two and four in terms of finite field period functions. As an application, we obtain new identities relating the Fourier coefficients of modular forms to special values of the finite field Appell series $F_1^p$ and $F_2^p$.

Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms

Abstract

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we construct two explicit families of eta-quotients, which we call the and functions, from the hypergeometric background. These and functions are constructed using the theory of weight Jacobi theta functions and their cubic analogues, respectively. Using these constructions, we then express the Fourier coefficients of certain Hecke eigenforms of weight two and four in terms of finite field period functions. As an application, we obtain new identities relating the Fourier coefficients of modular forms to special values of the finite field Appell series and .

Paper Structure

This paper contains 15 sections, 11 theorems, 76 equations, 7 tables.

Table of Contents

  1. Introduction and statement of results
  2. Appell series over finite fields and modular forms
  3. Preliminaries
  4. Euler's integral representation
  5. $p$-Adic preliminaries and Gross-Koblitz formula
  6. Commutative Formal Group Law (CFGL)
  7. The Explicit Hypergeometric Modularity Method (EHMM)
  8. Hypergeometric Character Sums and Extendability
  9. Construction of $\mathbb{K}_4$ and $\mathbb{K}_5$ functions
  10. The $\mathbb{K}_4$-functions
  11. The $\mathbb{K}_5$-functions
  12. The Galois Families for the $\mathbb{K}_4$ and $\mathbb{K}_5$ Functions
  13. Proof of Theorem \ref{['main-thm-K4']} and \ref{['main-thm-K5']}
  14. Proof of Theorem \ref{['rel-appell-modu-1']} and Theorem \ref{['rel-appell-modu-2']}
  15. Acknowledgement

Key Result

Theorem 1.1

Let $r \in \{1/2,1/3,1/4,1/6,1/8,1/12,1/24\}$. Let $f_{\mathrm{HD}_{\mathbb{K}_{4}\left(r\right)}}^{\sharp}$ be an explicit Hecke eigenform of weight four constructed from the EHMM. Then, for any prime ideal $\mathfrak{p}$ in $\mathbb{Z}[\zeta_{M}]$ lying above each fixed prime $p \equiv 1 \pmod{M:= The corresponding values of $\psi{(r)}(\mathfrak{p})$ and $f_{ \mathrm{HD}_{\mathbb{K}_{4}\left(r\r

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Proposition 2.2: beukers,Stienstra_Beukers
  • Theorem 2.3
  • Theorem 2.4: katz_galois1
  • Proposition 2.5
  • ...and 10 more