The Explicit Hypergeometric-Modularity Method

Acta Arithmetica, 2004

In a recent paper, Kaneko and Zagier studied a sequence of modular forms F k (z) which are solutions of a certain second order differential equation. They studied the polynomials e F k (j) = Y τ ∈H/Γ−{i,ω} (j − j(τ)) ord τ (F k) , where ω = e 2πi/3 and H/Γ is the usual fundamental domain of the action of SL 2 (Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e F p−1 (j) (mod p) is the nontrivial factor of the locus of supersingular j-invariants in characteristic p. Here we consider the irreducibility of these polynomials, and consider their Galois groups.

Transactions of the American Mathematical Society, 2007

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2 F 1 arises from a relation between modular curves, namely the covering of X 0 (3) by X 0 (9). In general, when 2 N 7, the N-fold cover of X 0 (N) by X 0 (N 2) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X 0 (6), X 0 (7) are of genus 1. Since their quotients X + 0 (6), X + 0 (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.

Journal of the Australian Mathematical Society, 2018

We record $\binom{42}{2}+\binom{23}{2}+\binom{13}{2}=1192$ functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulas for $1/\unicode[STIX]{x1D70B}$ and in the computation of mathematical constants.

Contemporary Mathematics, 1994

There are four values of s for which the hypergeometric function 2 F 1 ( 1 2 − s, 1 2 + s; 1; ·) can be parametrized in terms of modular forms; namely, s = 0, 1 3 , 1 4 , 1 6 . For the classical s = 0 case, the parametrization is in terms of the Jacobian theta functions θ 3 (q), θ 4 (q) and is related to the arithmetic-geometric mean iteration of Gauss and Legendre. Analogues of the arithmetic-geometric mean are given for the remaining cases. The case s = 1 6 and its relationship to the work of Ramanujan is highlighted. The work presented includes various pieces of joint work with combinations of the following:

Acta Arithmetica, 2010

Gauss's hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form

2016

Journal of Number Theory, 2012

In this article using the theory of Eisenstein series, we give the complete evaluation of two Gauss hypergeometric functions. Moreover we evaluate the modulus of each of these functions and the values of the functions in terms of the complete elliptic integral of the first kind K. As application we give way of how to evaluate the parameters in a closed-well posed form, of a general Ramanujan type 1/π formula. The result is a formula of 110 digits per term.

International Mathematics Research Notices

We establish a simple inductive formula for the trace $${\hbox{ Tr }}_{k}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right),p\right)$$ of the p th Hecke operator on the space $${S}_{k}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right)\right)$$ of newforms of level 8 and weight k in terms of the values of 3 F 2 -hypergeometric functions over the finite field F p . Using this formula when k = 6, we prove a conjecture of Koike relating $${\hbox{ Tr }}_{6}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right),p\right)$$ to the values 6 F 5 (1) p and 4 F 3 (1) p . Furthermore, we find new congruences between $${\hbox{ Tr }}_{k}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right),p\right)$$ and generalized Apéry numbers.

The Ramanujan Journal, 2014

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the threepunctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.

2008

The classical theory of elliptic modular equations is refor mulated and extended, and many new rationally parametrized modular equations are dis covered. Each arises in the context of a family of elliptic curves attached to a genus-zero congrue nce subgroupΓ0(N), as an algebraic transformation of elliptic curve periods, which are parame triz d by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equat ion, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algeb r ic transformation formulas for special functions. The ones for N = 4, 3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3, 4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.