arXiv subjects

Esme Rosen

Publications and source records attributed to Esme Rosen.

3 recordsLinked to original sources

arXiv2026

The Explicit Hypergeometric Modularity Method III

We refine the Explicit Hypergeometric Modularity Method (EHMM) and develop a variant that applies to a broader class of hypergeometric data. As an application, we establish the modularity of hypergeometric Galois representations arising from length-$4$ data that are not necessarily defined over $\mathbb{Q}$. We also use this method to give explicit constructions of nine modular forms associated with hypergeometric rigid Calabi-Yau threefolds conjectured to be modular by Rodriguez-Villegas. The modularity of these threefolds was first proved by Long-Tu-Yui-Zudilin using a different approach based on the Faltings-Serre method. Moreover, the explicit nature of the method makes it well suited to the computation of special $L$-values of the associated modular forms.

math.NT↗

arXiv2025

$L$-values of certain weight 3 Modular Forms and Transformations of Hypergeometric Series

Recently, Allen, Grove, Long, and Tu proposed an explicit Hypergeometric-Modularity method which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3 modular forms. Among other properties their process shows that the $L$-value of such a modular form at 1 is an explicit multiple of a ${}_3F_2(1)$ hypergeometric series. Using the framework of a finite Coxeter group governing the invariance group of normalized ${}_3F_2(1)$ series, this paper fully classifies and describes the possible Hecke eigenforms whose $L$-values that can be obtained using this method. In addition, we determine when these modular forms differ by twist of a finite-order character using the perspective of hypergeometric functions. As one application, we reinterpret a classical identity of hypergeometric series as a formula involving $L$-values of two Hecke eigenforms that differ by a twist.

math.NT↗

arXiv2025

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

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.

math.NT↗