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

Abstract

Abstract. We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can be often read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of p and from Weil's general bounds |a(p)| ≤ 2p To Noriko Yui, with wishes to count more points on algebraic varieties rather than years! A prototypeIn [32] L. van Hamme stated some supercongruence analogues of Ramanujan's formulas. The very last observation on van Hamme's list, Conjecture (M.2) (stated here in an equivalent form), does not seem to be linked to a known formula though:where a(n) denote the Fourier coefficients of the unique cusp (eigen) form of weight 4 on Γ 0 (8),

Interpolated Sequences and Critical L-Values of Modular Forms

“…We first prove case D in detail, then briefly indicate how to establish cases A, B, C and E. We conclude with a sketch of case F . We note that the relation of the hypergeometric series, which arise for sequences A, C, D, and the corresponding L-values already appears in [42].…”

“…Does there exist a version of Theorem 3.2 in this case? Fifth, Zudilin [42] recently considered periods of certain instances of rigid Calabi-Yau manifolds, which are expressed in terms of hypergeometric functions. In these instances, he conjecturally indicated a relation between special bases of the hypergeometric differential equations and all critical L-values of the corresponding modular forms (these relations include those that we observed during the proof of Theorem 2.2).…”

Distribution of values of Gaussian hypergeometric functions

“…These properties include transformation laws, explicit evaluations, and contiguous relations. These functions have played central roles in the study of combinatorial supercongruences [1,3,36,43,46,47,51,54,55,56,57,58], Dwork hypersurfaces [9,45], Galois representations [40,41], L-functions of elliptic curves [6,10,11,25,39,44,52,60,63], hyperelliptic curves [7,8], K3 surfaces [4,19,52], Calabi-Yau threefolds [2,3,64], the Eichler-Selberg trace formula [24,25,26,27,38,48,58,59], among other topics.…”

Special Hypergeometric Motives and Their L-Functions: Asai Recognition

“…Motives attached to 3 F 2 hypergeometric functions are reasonably well understood (see e.g. Zudilin [40,Observation 4]), and we review them briefly in Section 2. By contrast, the 5 F 4 motives associated with similar formulas had not been linked explicitly to modular forms.…”