Sequences, modular forms and cellular integrals

Abstract

It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

Apéry-like numbers and families of newforms with complex multiplication

“…Applying the methods used to prove Theorems 1.4 and 1.8, to the results in [13], gives a slight enhancement of (1.2). Theorem 1.9.…”

“…More recently, Brown [7] introduced cellular integrals which can be expressed as Q-linear combinations of multiple zeta values, and include Beukers integrals as a special cases. In [13], the authors, of whom the second author of this paper is one, examine sequences arising from the coefficients in these linear combinations. They show that all powers of the Apéry numbers are among these sequences and that they too satisfy congruence relations with Fourier coefficients of modular forms which have complex multiplication by Q( √ −1), similar to (1.2).…”

Interpolated Sequences and Critical L-Values of Modular Forms

“…Remark 2.5. Let us illustrate how one can algorithmically derive and prove (27). Let D(x, k) be the summand in the sum defining A(x).…”

“…For example, if N = 5, then σ 5 = (1, 3, 5, 2, 4) is the unique convergent permutation, I σ5 (n) recovers Beukers' integral for ζ(2) [8] and the leading coefficients A σ5 (n) are the Apéry numbers C D (n) in (6). In [27], an explicit family σ N of convergent configurations for odd N ≥ 5 is constructed such that the leading coefficients A σN (n) are powers of the Apéry numbers C D (n), that is,…”

“…These are linear forms in multiple zeta values and their leading coefficients A σ (n) are generalizations of the Apéry numbers. McCarthy and the authors [27] proved that, for a certain infinite family of these cellular integrals, the leading coefficients A σ (n) satisfy congruences like (4) with Fourier coefficients of modular forms f k (τ ) of odd weight k ≥ 3. In Section 3, we review these facts and prove an analogue of Zagier's evaluation (2) for all of these sequences.…”