Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms

Published 3 Apr 2026 in math.NT | (2604.02723v1)

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 $F_2^p$.

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Summary

The paper constructs explicit eta-quotient families (𝕂₄ and 𝕂₅) from hypergeometric data to study weight 2 and 4 Hecke eigenforms.
It employs the Explicit Hypergeometric Modularity Method to relate finite field character sums to modular form Fourier coefficients through clear identities.
The rigorous theoretical and numerical results extend modularity to new hypergeometric and Appell series contexts, deepening insights into arithmetic geometry.

Explicit Hypergeometric Modularity for Weight Two and Four Hecke Eigenforms

Background and Motivation

This work leverages the Explicit Hypergeometric Modularity Method (EHMM), recently developed by Allen et al., to systematically connect hypergeometric character sums—specifically finite field analogues—and modular forms arising from Hecke eigenforms. The classical theory of Gaussian hypergeometric series and their period functions provides a rich framework for relating Fourier coefficients of modular forms to explicit character sums, extending established results in both dimension two and three to new classes in higher weights.

The primary advancement of this paper lies in constructing two explicit families of modular eta-quotients, denoted as $\mathbb{K}_4$ and $\mathbb{K}_5$ , arising naturally from hypergeometric data. These constructions, based on weight $1/2$ Jacobi theta functions and their cubic analogues, enable explicit formulas for Fourier coefficients of Hecke eigenforms of weights two and four via finite field period functions, with direct applications to identities involving the Appell series $F_1^p$ and $F_2^p$ .

Hypergeometric Modular Construction and Eta-Quotient Families

Hypergeometric Data and Finite Field Analogues

Starting from classical hypergeometric data $\mathrm{HD} = \{\alpha, \beta\}$ , where $\alpha$ and $\beta$ are multisets of rational parameters and $M$ is their least common denominator, the paper defines finite field analogues for associated hypergeometric and period functions using multiplicative characters derived from rational exponents. These character sums, as defined by Greene and Fuselier et al., encode crucial arithmetic information on modular forms.

For a fixed hypergeometric datum, these character sums are tailored to express traces of Frobenius associated to eigenforms, following the explicit realization via Katz’s Galois representation construction.

Construction of $\mathbb{K}_4$ and $\mathbb{K}_5$ 0 Eta-Quotients

The $\mathbb{K}_5$ 1 family is constructed for weight four via the data:

$\mathbb{K}_5$ 2

where $\mathbb{K}_5$ 3 ranges over $\mathbb{K}_5$ 4. These eta-quotients are explicitly linked to Jacobi theta functions and their modular Hauptmodul transformations, ensuring congruence holomorphic cuspforms with precise levels and characters.

Similarly, the $\mathbb{K}_5$ 5 family for weight two is built from:

$\mathbb{K}_5$ 6

with $\mathbb{K}_5$ 7 in $\mathbb{K}_5$ 8. The cubic analogues of Jacobi theta functions underpin the explicit eta-quotient modular forms.

For both families, explicit computations of Hecke operator actions on these eta-quotients permit the construction of normalized Hecke eigenforms. The Galois cases are characterized by conjugacy relations among parameters and their completion to Hecke orbits.

Explicit Modularity Identities and Trace Formulas

Hypergeometric Modularity Results

The main theorems establish explicit identities linking finite field period functions (hypergeometric character sums) and Fourier coefficients of normalized Hecke eigenforms, for primes $\mathbb{K}_5$ 9:

For the weight four $1/2$0 family:

$1/2$1

where $1/2$2 and $1/2$3 are explicitly determined.

For the weight two $1/2$4 family:

$1/2$5

providing an explicit expansion in terms of hypergeometric character sums.

These identities, proven using EHMM, CFGL, and the Gross-Koblitz formula, are valid for all $1/2$6 (with smaller primes verified computationally), and involve explicit normalization constants and characters determined by the analytic and arithmetic properties of the eta-quotients.

Appell Series and Further Applications

The study extends to finite field analogues of the Appell series $1/2$7 and $1/2$8, providing new modularity identities—previously unavailable except for $1/2$9—connecting Fourier coefficients of weight two modular forms to special values of these double series:

Relations such as:

$F_1^p$ 0

and analogous expressions for $F_1^p$ 1 when $F_1^p$ 2, establish explicit formulae for traces of Hecke operators in terms of period values of finite field Appell series.

Numerical Results and Contradictory Assertions

Strong numerical claims are present throughout:

Integer-valued hypergeometric period functions for specified $F_1^p$ 3 and explicit correspondence with LMFDB-labeled modular forms.
Bounds from Deligne and explicit norm computations ensure that congruence identities hold for all sufficiently large primes; for small primes, direct computation resolves any ambiguities.

Contradictory to prior expectations, explicit modularity results are provided for the Appell $F_1^p$ 4 and $F_1^p$ 5 series, resolving gaps in the literature where previous works only covered $F_1^p$ 6.

Implications and Future Directions

This explicit hypergeometric modularity framework confirms and extends the reach of modular character sum identities in higher weights, systematically linking families of eta-quotients—modular forms arising from hypergeometric data—to period functions and Appell series over finite fields. The approach facilitates further arithmetic exploration of congruence relations, $F_1^p$ 7-value identities, and Galois representations associated to modular forms of diverse type.

Expected future directions include:

Extension of the EHMM to even higher-weight or noncongruence modular forms, exploiting other types of period functions.
Systematic classification of Galois families for new modular eta-quotients and their explicit Hecke orbits.
Deeper incorporation of Appell and multivariate hypergeometric series in the landscape of modularity, potentially linking to point-counting on higher-dimensional varieties.

Conclusion

This research rigorously advances explicit modularity results by constructing new eta-quotient families $F_1^p$ 8 and $F_1^p$ 9, establishing sharp arithmetic connections between their Fourier coefficients and finite field hypergeometric character sums. It also derives new identities linking weight two modular forms to values of Appell series over finite fields. The technical machinery employed, notably the EHMM, enables precise congruence and Galois-theoretic assertions, widening the landscape of explicit modularity and laying the groundwork for further exploration into modular forms, hypergeometric functions, and arithmetic geometry.