On p-adic modular forms and the Bloch–Okounkov theorem

Michael J Griffin,Sarah Trebat-Leder,Marie Jameson

doi:10.1186/s40687-016-0055-z

Abstract

Bloch–Okounkov studied certain functions on partitions f called shifted symmetric polynomials. They showed that certain q-series arising from these functions (the so-called q-brackets \(\left _q\)) are quasimodular forms. We revisit a family of such functions, denoted \(Q_k\), and study the p-adic properties of their q-brackets. To do this, we define regularized versions \(Q_k^{(p)}\) for primes p. We also use Jacobi forms to show that the \(\left _q\) are quasimodular and find explicit expressions for them in terms of the \(\left _q\).

Highlights

Introduction and statement of resultsIn [10], Serre introduced the theory of p-adic modular forms, which are p-adic limits of compatible families of q-expansions of classical level one modular forms
Mk, where k is a positive even integer, τ ∈ H, q = e2πiτ, Bk is the kth Bernoulli number, σk−1(n) is the sum of the k − 1 powers of the divisors of n, and Mk is the space of weight k quasimodular forms
A result Bloch and Okounkov [3] gives that for a large class of functions f : P → Q, called shifted symmetric polynomials, the q-series f q is a quasimodular form on the full modular group

Summary

Introduction and statement of results

In [10], Serre introduced the theory of p-adic modular forms, which are p-adic limits of compatible families of q-expansions of classical level one modular forms. Dk−1, and Bk(p) := (1 − pk−1)Bk. In order to find congruences for these regularized Eisenstein series, we recall Euler’s theorem, which says that if (a, n) = 1 aφ(n) ≡ 1 (mod n), where φ(n) := #{k ∈ Z, 1 ≤ k ≤ n, (n, k) = 1}. For any function f : P → Q, we define the “q-bracket of f ” to be the following formal power series obtained by “averaging”:. A result Bloch and Okounkov [3] gives that for a large class of functions f : P → Q, called shifted symmetric polynomials, the q-series f q is a quasimodular form on the full modular group. The Bloch–Okounkov Theorem states that for f ∈ R homogeneous of grading k, f q is a quasimodular form of weight k on the full modular group.

Preliminary definitions

Congruences and p-adic modular forms

Jacobi forms

Showing quasimodularity