Irrationality of Certain p - adic Periods for Small p

Apéry’s proof [13] of the irrationality of ζ(3) is now over 25 years old, and it is perhaps surprising that his methods have not yielded any significant new results (although further progress has been made on the irrationality of zeta values [1, 14]). Shortly after the initial proof, Beukers produced two elegant reinterpretations of Apéry’s arguments; the first using iterated integrals and Legendre polynomials [2], and the second using modular forms [3]. It is this second argument that we will apply to study the irrationality of certain p-adic periods, in particular, the p-adic analogues of ζ(3) and Catalan’s constant for small p. To relate certain classical periods to modular forms, Beukers considers various integrals of holomorphic modular forms that themselves satisfy certain functional equations (analogous to the functional equation for the nonholomorphic Eisenstein series of weight two). That these integrals satisfy functional equations is a consequence of the theory of Eichler integrals. The periods arise as coefficients of the associated period polynomials. In our setting, these auxiliary functional equations are replaced by the notion of overconvergent p-adic modular forms [6, 7, 10]. In this guise, our p-adic periods will occur as coefficients of overconvergent Eisenstein series of negative integral weight. These p-adic periods are equal to special values of Kubota–Leopoldt p-adic L-functions. Thus ζ(3) is replaced by ζp(3) = Lp(3, id) and Catalan’s constant