Explicit Modularity of Hypergeometric Motives

Abstract: The Modularity Theorem states that given an elliptic curve one can find an associated modular form. One of the more striking aspects of the Modularity Theorem is the variety of seemingly unrelated ways in which the relationship between the elliptic curve and the modular form can be stated. For this talk, the primary formulations of modularity we will be interested in are the equality of elliptic and modular $L$-functions, equality between the number of points on the elliptic curve mod $p$ with the Fourier coefficients of the modular form, and finally an isomorphism between elliptic and modular Galois representations. Each of these connections can be made explicit by expressing both sides in terms of hypergeometric functions (over $\mathbb{C}$), hypergeometric character sums (over $\mathbb{F}_p$), and hypergeometric Galois representations (over $\mathbb{Q}_\ell)$. More generally, each of these connections correspond to De Rham, crystalline, and étale realizations of hypergeometric motives. We discuss recent and upcoming work with Grove, Long, and Tu using these hypergeometric perspectives towards understanding generalizations of the Modularity Theorem for these hypergeometric motives.

number theory

Audience: researchers in the topic

Five College Number Theory Seminar