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Modular Forms and Certain ${}_2F_1(1)$ Hypergeometric Series

Esme Rosen

TL;DR

The paper develops a hypergeometric modularity framework to produce a CM-weight $2$ family of Hecke eigenforms from ${}_2F_1(1)$ data, and derives exact central $L$-values in terms of beta values and Chowla–Selberg periods. It shows Fourier coefficients are expressible via Jacobi sums, establishing a motivic link through Jacobi motives and Fermat-curve quotients, with explicit eta-product realizations and a detailed description of the Hecke action. It also provides explicit Galois-representations descriptions for the CM forms, along with comprehensive tables of exact $L$-values and twisting relations, together with concrete examples illustrating connections to BSD-type conjectures. The approach extends previous hypergeometric-modularity work by exploiting the CM structure to obtain precise $L$-values, while delivering a clean CM-setup via Jacobi motives that yields new CM modular forms and tractable arithmetic data. Overall, the work delivers explicit CM weight $2$ modular forms tied to hypergeometric data, explicit $L$-values in terms of Chowla–Selberg periods, and a framework connecting Jacobi sums, eta-quotients, and Fermat-curve motives to modularity.

Abstract

Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of alternative bases to compute the exact central $L$-value of these Hecke eigenforms in terms of special beta values. We also show the integral Fourier coefficients can be written in terms of Jacobi sums, reflecting a motivic relation between the hypergeometric series and the modular forms.

Modular Forms and Certain ${}_2F_1(1)$ Hypergeometric Series

TL;DR

The paper develops a hypergeometric modularity framework to produce a CM-weight family of Hecke eigenforms from data, and derives exact central -values in terms of beta values and Chowla–Selberg periods. It shows Fourier coefficients are expressible via Jacobi sums, establishing a motivic link through Jacobi motives and Fermat-curve quotients, with explicit eta-product realizations and a detailed description of the Hecke action. It also provides explicit Galois-representations descriptions for the CM forms, along with comprehensive tables of exact -values and twisting relations, together with concrete examples illustrating connections to BSD-type conjectures. The approach extends previous hypergeometric-modularity work by exploiting the CM structure to obtain precise -values, while delivering a clean CM-setup via Jacobi motives that yields new CM modular forms and tractable arithmetic data. Overall, the work delivers explicit CM weight modular forms tied to hypergeometric data, explicit -values in terms of Chowla–Selberg periods, and a framework connecting Jacobi sums, eta-quotients, and Fermat-curve motives to modularity.

Abstract

Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of hypergeometric series and Ramanujan's theory of alternative bases to compute the exact central -value of these Hecke eigenforms in terms of special beta values. We also show the integral Fourier coefficients can be written in terms of Jacobi sums, reflecting a motivic relation between the hypergeometric series and the modular forms.

Paper Structure

This paper contains 11 sections, 12 theorems, 46 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hecke Operators
  4. Jacobi Motives
  5. The Action of the Hecke Operators
  6. Galois Representations
  7. Special L-values
  8. Computing L-values
  9. Twisting and Relations between L-values
  10. Tables of L-values
  11. Appendix: Construction of the Hecke Eigenforms using Eta Products

Key Result

Theorem 1.1

Let $M:=\text{lcd}(r,s,1/2)$ be the least common denominator of $r$, $s$, and $1/2$, and $G_M$ be the absolute Galois group of $\mathbb{Q}(\zeta_M)$. Further, take $\Omega_{-D}$ to be the Chowla-Selberg period of the imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$. Then for each pair $(r,s)\in \ma A list of exact $L$-values is provided in Section lvalues.

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 11 more