Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Traces of Hecke Operators via Hypergeometric Character Sums

Jerome W. Hoffman, Wen-Ching Winnie Li, Ling Long, Fang-Ting Tu

TL;DR

This work addresses the computation of traces of Hecke operators on cusp forms for arithmetic triangle groups by embedding the problem into the geometry of automorphic and hypergeometric sheaves. The authors develop a unified framework linking $-\mathrm{Tr}(T_p|S_{k+2}(\Gamma))$ with Frobenius traces on $H^1(X_{\Gamma},V^k(\Gamma)_{\ell})$, and express these traces explicitly as hypergeometric character sums $H_p(HD(\Gamma);\cdot)$, including CM and cusp contributions. The main contributions include a comprehensive trace formula for several groups (including $(2,4,6)$) and a detailed analysis of singular fibers, CM points, and Atkin–Lehner involutions, plus verifications in low-weight cases and extensions to other subgroups. The results illuminate deep connections between modular/ Shimura forms, hypergeometric motives, and CM theory, offering practical means to compute eigenvalues and revealing new identities for hypergeometric sums with potential applications to special values of hypergeometric functions and Sato–Tate-type questions.

Abstract

In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite fields, a theory developed largely by Greene, Katz, Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach, in contrast to the previous works, is uniform and more geometric, and it works equally well for forms on elliptic modular curves and Shimura curves. The same method can be applied to obtain eigenvalues of Hecke operators as well.

Traces of Hecke Operators via Hypergeometric Character Sums

TL;DR

This work addresses the computation of traces of Hecke operators on cusp forms for arithmetic triangle groups by embedding the problem into the geometry of automorphic and hypergeometric sheaves. The authors develop a unified framework linking with Frobenius traces on , and express these traces explicitly as hypergeometric character sums , including CM and cusp contributions. The main contributions include a comprehensive trace formula for several groups (including ) and a detailed analysis of singular fibers, CM points, and Atkin–Lehner involutions, plus verifications in low-weight cases and extensions to other subgroups. The results illuminate deep connections between modular/ Shimura forms, hypergeometric motives, and CM theory, offering practical means to compute eigenvalues and revealing new identities for hypergeometric sums with potential applications to special values of hypergeometric functions and Sato–Tate-type questions.

Abstract

In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite fields, a theory developed largely by Greene, Katz, Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach, in contrast to the previous works, is uniform and more geometric, and it works equally well for forms on elliptic modular curves and Shimura curves. The same method can be applied to obtain eigenvalues of Hecke operators as well.
Paper Structure (30 sections, 37 theorems, 212 equations, 1 figure, 7 tables)

This paper contains 30 sections, 37 theorems, 212 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Automorphic sheaves
  5. Complex automorphic sheaves
  6. $\ell$-adic automorphic sheaves
  7. Groups
  8. Hypergeometric functions over $\mathbf{C}$
  9. A motivating example
  10. The groups to be considered
  11. Isomorphisms using rigidity
  12. Comparison Theorem
  13. The $\ell$-adic sheaves isomorphism
  14. Hypergeometric character sums
  15. Hypergeometric functions over finite fields
  16. ...and 15 more sections

Key Result

Theorem 1

For $\Gamma = ~(2,4,6),~(2,\infty,\infty),(2,3,\infty),(2,4,\infty),(2,6,\infty)$, the table below describes the hypergeometric datum $HD(\Gamma)=\{\alpha(\Gamma),\beta(\Gamma)\}$ and the choice of a generator $\lambda=\lambda(\Gamma)$ of the field of $\mathbf{Q}$-rational functions on $X_\Gamma$ by Given an even integer $k\ge 2$ and a fixed prime $\ell$, the terms on the right-hand side of (E:mot

Theorems & Definitions (57)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Theorem 3: Deligne Deligne-mf, Ohta Ohta82
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 47 more