Higher-order Maass forms

Abstract

The spaces of Maass forms of even weight and of arbitrary order are studied. It is shown that, if we allow exponential growth at the cusps, these spaces are as large as algebraic restrictions allow. These results also apply to higher-order holomorphic forms of even weight.

Arithmetic statistics of modular symbols

“…From now on we assume α to be harmonic. We remark that L(ε) as a function of ε is a polynomial of degree 2, and that L (1) and L (2) defined in (4.1) and (4.2) are the first and second derivative of L(ε) at ε = 0. The eigenvalue equation for E a (z, s, ε) and (2.13) imply that…”

“…We will now describe the full singular part of L (2) ab (s, 0, 0) at s = 1. We denote the constant term in the Laurent expansion of E a (z, s) by B a (z), i.e.…”

“…It follows that the singular part of (2) ab (s) equals the singular part of (2s − 1) (z). It follows that the singular part of the third term of (6.8) equals the singular part of…”

“…We remark that Theorem 6.3 allows us to write the singular expansion of L (2) ab (s, 0, 0) exclusively in terms of data of Rankin-Selberg integrals and periods. Indeed, writing…”

Harmonic lifts of modular forms

“…has the transformation behavior H(γτ) = H(τ) + λ(γ) for some group homomorphism λ : Γ com → C. The function C(z) = −1 4π H satisfies ξ 0 C = η4 . In §4.3.1 in [6] we see that there can be found a linear combination M of the holomorphic…”