Finite hypergeometric functions

Abstract

We show that values of finite hypergeometric functions defined over Q correspond to point counting results on explicit varieties defined over finite fields.

Hypergeometric decomposition of symmetric K3 quartic pencils

“…Proof of Proposition 3.6.2. If q ≡ 1 (mod 4) then, by Lemmas B.3.5 and B.3.16, we have that [BCM15] again to shift parameters, then we see that in fact all of the Picard-Fuchs equations satisfied by the non-holomorphic periods correspond to this same hypergeometric motive over Q.…”

“…We associate a field of definition K α α α,β β β to α α α, β β β, which is an explicitly given finite abelian extension of Q. For certain prime powers q and t ∈ F q , there is a finite field hypergeometric sum H q (α α α; β β β | t) ∈ K α α α,β β β defined by Katz [Kat90] as a finite field analogue of the complex hypergeometric function, normalized by McCarthy [McC12b], extended by Beukers-Cohen-Mellit [BCM15], and pursued by many authors: see section 3.1 for the definition and further discussion, and section 3.2 for an extension of this definition. We package together the exponential generating series associated to these hypergeometric sums into an L-series L S (H(α α α; β β β | t), s): see section 4.1 for further notation.…”

“…Dwork studied the now-eponymous Dwork pencil [Dwo69, §6j, p. 73], and Candelas-de la Ossa-Rodríguez-Villegas considered the factorization of the zeta function for the Dwork pencil of Calabi-Yau threefolds in [CDRV00,CDRV01], linking physical and mathematical approaches. More recently, given a finite-field hypergeometric function defined over Q, Beukers-Cohen-Mellit [BCM15] construct a variety whose trace of Frobenius is equal to the finite field hypergeometric sum up to certain trivial factors.…”

Fields of Definition of Finite Hypergeometric Functions

“…One of the obstacles in the definition of finite hypergeometric sums over Q is Assumption 1.1 which has to be made on q, whereas one has the impression that such sums can be defined for any q relatively prime with the common denominator of the α i , β j . This is resolved in [1,Thm 1.3] by an extension of the definition of hypergeometric sum. The idea is to apply the theorem of Hasse-Davenport to the products of Gauss sums which occur in the coefficients of the hypergeometric sum.…”

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

“…Note that the identity (4) is the missing non-p-adic counterpart (M.1) of Conjecture (M.2) from [32]; the latest edition of van Hamme's list can be found in [31] together with the details about proofs. One of the principal results in [1] is a summation formula for Greene's hypergeometric function, which serves as a finite-field analogue of the classical hypergeometric series given in (4). Curiously enough, R. Evans in his review [7] of [1] mentions that no summation formula is known for this 4 F 3 -value in (4); the evaluation (4) established in [23,34] thus fills in this gap in the hypergeometric literature.…”