Irrationality of certain p-adic periods for small p

Details Irrationality of certain p-adic periods for small p Irrationality of certain p-adic periods for small p

2004-08-16

arxiv.org/abs/math/0408214v1

Mathematics

Number Theory

Preprint

Scientific paper

Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain functional equations arising from Eichler integrals are replaced by the notion of overconvergent p-adic modular forms, and the periods themselves arise not as coefficients of period polynomials but as constant terms of p-adic Eisenstein series. We prove that the analogue of zeta(3) is irrational for p = 2 and 3, as well as the 2-adic analogue of Catalan's constant.

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