Related papers: Hypergeometric functions over F_p and relations to…
We introduce hypergeometric functions related to projective Schur functions $Q_{\lambda}$ and describe their properties. Linear equations, integral representations and Pfaffian representations are obtained. These hypergeometric functions…
We specialize the Eichler-Selberg trace formula to obtain trace formulas for the prime-to-level Hecke action on cusp forms for certain congruence groups of arbitrary level. As a consequence, we determine the asymptotic in the prime p of the…
We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the…
We apply transformer models and feedforward neural networks to predict Frobenius traces $a_p$ from elliptic curves given other traces $a_q$. We train further models to predict $a_p \bmod 2$ from $a_q \bmod 2$, and cross-analysis such as…
We describe the action of the Dwork-Frobenius operator on certain $A$-hypergeometric series. As a consequence, we obtain an integrality result for the coefficients of those series. This implies an integrality result for classical…
We define "values" of the elliptic modular j-function at real quadratic irrationalities by using Hecke's hyperbolic Fourier expansions, and present some observations based on numerical experiments.
In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular,…
Inspired by a result of Manin, we study the relationship between certain period integrals and the trace of Frobenius of genus 3 generalized Legendre curves. We show that both of these properties can be computed in terms of "matching"…
We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.
The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss…
We apply the method established in our previous work to derive a Chudnovsky-Ramanujan type formula for the Legendre family of elliptic curves. As a result, we prove two identities for $1/\pi$ in terms of hypergeometric functions.
For a fixed positive integer $e$, we describe an algorithm for computing, for all primes $p \leq X$, the mod-$p^e$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive over $\mathbb{Q}$ in time quasilinear in $X$.…
We give an expression for number of points for the family of Dwork K3 surfaces $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene's finite field…
Finite hypergeometric functions are complex valued functions on finite fields which are the analogue of the classical analytic hypergeometric functions. From the work of N.M.Katz it follows that their values are traces of Frobenius on…
This is a report on recent work, with Wen-Ching Winnie Li and Ling Long. In that work explicit formulas are given, involving hypergeometric character sums, for the traces of Hecke operators $T_p$ acting spaces of cusp forms $S_k(\Gamma)$ of…
We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of…
Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely…
We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials considered by S.Milne \cite{Milne}. For $q=1$ these functions are also known as hypergeometric functions of matrix argument…
We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These…
We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of…