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Special $L$-values of certain CM weight three Hecke eigenforms

Paresh Arora, Koustav Mondal, Akio Nakagawa, Fang-Ting Tu

TL;DR

The paper develops a framework to evaluate special values of $L$-functions attached to CM weight-3 Hecke eigenforms by connecting Ramanujan's alternative bases of modular forms to hypergeometric series via modular hypergeometric representations, using CM elliptic curves over real quadratic fields. It constructs hypergeometric Galois representations associated to CM data and proves their modularity, expressing $L(f_{d,t},2)$ explicitly in terms of ${}_3F_{2}$ evaluations and Chowla–Selberg periods; a complete classification of these representations is provided. For $d ext{ in }\{2,3,4,6ig ext{,} extrm{t}ig ext{)}$, the paper lists all CM-valued data $(d,t)$, rational constants $C_{d,t}$, and CM points $ au_d$ that realize the modularity, with explicit L-value formulas such as $C_{d,t} imes L(f_{d,t},2)= obreak rac{ ext{constant}}{ } igl[{}_3F_{2}( frac{1}{2}, frac{1}{d}, frac{d-1}{d};1,1;t)igr]$ and connections to $j( au)$-invariants. By intertwining hypergeometric identities, CM theory, and modularity, the work extends the CM-structure of higher-weight motives and yields concrete evaluations of higher-weight CM $L$-values, with potential extensions to higher weight and Petersson inner products. The results illuminate aspects of the Langlands program for hypergeometric motives and provide practical, CM-based evaluations of weight-3 CM forms via hypergeometric data.

Abstract

Ramanujan's theories of elliptic functions to alternative bases for modular forms connect hypergeometric series with modular forms and have led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of $L$-function of certain CM cusp forms to Ramanujan's alternative bases via the modularity of hypergeometric Galois representations arising from CM elliptic curves over real quadratic fields. We also give a complete classification of these representations.

Special $L$-values of certain CM weight three Hecke eigenforms

TL;DR

The paper develops a framework to evaluate special values of -functions attached to CM weight-3 Hecke eigenforms by connecting Ramanujan's alternative bases of modular forms to hypergeometric series via modular hypergeometric representations, using CM elliptic curves over real quadratic fields. It constructs hypergeometric Galois representations associated to CM data and proves their modularity, expressing explicitly in terms of evaluations and Chowla–Selberg periods; a complete classification of these representations is provided. For , the paper lists all CM-valued data , rational constants , and CM points that realize the modularity, with explicit L-value formulas such as and connections to -invariants. By intertwining hypergeometric identities, CM theory, and modularity, the work extends the CM-structure of higher-weight motives and yields concrete evaluations of higher-weight CM -values, with potential extensions to higher weight and Petersson inner products. The results illuminate aspects of the Langlands program for hypergeometric motives and provide practical, CM-based evaluations of weight-3 CM forms via hypergeometric data.

Abstract

Ramanujan's theories of elliptic functions to alternative bases for modular forms connect hypergeometric series with modular forms and have led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of -function of certain CM cusp forms to Ramanujan's alternative bases via the modularity of hypergeometric Galois representations arising from CM elliptic curves over real quadratic fields. We also give a complete classification of these representations.
Paper Structure (23 sections, 22 theorems, 173 equations, 1 figure, 3 tables)

This paper contains 23 sections, 22 theorems, 173 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Hypergeometric functions
  4. Hypergeometric transformation formulae
  5. Hypergeometric elliptic curves
  6. Modular Forms
  7. Eisenstein series
  8. Action of Fricke involution
  9. Twisting modular forms
  10. Modular equations and special values
  11. Relation between Hypergeometric functions and Modular forms
  12. Galois Representations
  13. Hecke Characters and Hecke L-functions
  14. Hypergeometric Galois representations
  15. Case 1: $z \in \mathbb{Q}$ and $z\neq 1/2$
  16. ...and 8 more sections

Key Result

Theorem 1.1

For each $d\in\{2,3,4,6\}$, let $t \in \mathbb{Q}$ and $z=\frac{1-\sqrt{1-t}}{2}$ with $\mathbb{Q}(z) \subset \mathbb{R}$ such that the elliptic curve $\widetilde{E}_d\left(z\right)$ given in eq:hgellipticcurves has complex multiplication. Then there exists a unique Hecke eigenform $f_{d,t}$ of weig where $(a)_0=1$ and $(a)_n=a(a+1)\cdots(a+n-1)$, and the $f_{d,t}$'s are listed in LMFDB labels LMF

Figures (1)

  • Figure 1: Fundamental domains for (A) $\Gamma_0(2)^+$, (B) $\Gamma_0(3)^+$, (C) $\Gamma_0(1)$ and (D) $\Gamma_0(2)$ respectively.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Example 1.4
  • Remark 1.5
  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 3.1
  • Proposition 3.2
  • ...and 37 more