On a supercongruence conjecture of Rodriguez-Villegas
Dermot McCarthy

TL;DR
This paper proves a significant supercongruence conjecture linking hypergeometric series and modular form coefficients, advancing understanding of arithmetic properties of Calabi-Yau manifolds.
Contribution
It provides a proof for a key supercongruence conjecture connecting hypergeometric series with modular forms, a problem previously unresolved.
Findings
Confirmed a supercongruence conjecture for a specific hypergeometric series
Established a link between hypergeometric series values and modular form coefficients
Enhanced understanding of arithmetic properties of Calabi-Yau manifolds
Abstract
In examining the relationship between the number of points over on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the -th Fourier coefficient of a modular form.
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Combinatorial Mathematics
