Steinberg homology, modular forms, and real quadratic fields

Perfect unary forms over real quadratic fields

Let F = Q( √ d) be a real quadratic field with ring of integers O. In this paper we analyze the number hd of GL1(O)orbits of homothety classes of perfect unary forms over F as a function of d. We compute hd exactly for square-free d ≤ 200000. By relating perfect forms to continued fractions, we give bounds on hd and address some questions raised by Watanabe, Yano, and Hayashi.

Modular Forms and Elliptic Curves over Imaginary Quadratic Fields

`-adic Representations Associated to Modular Forms over Imaginary Quadratic Fields

Let π be a regular algebraic cuspidal automorphic representation of GL2 over an imaginary quadratic number field K, and let ` be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible `-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explici...

Quadratic Minima and Modular Forms

1991 Mathematics Subject Classi cation: 11F11, 11E20.

Modular Invariants for Real Quadratic Fields and Kloosterman Sums

We investigate the asymptotic distribution of integrals of the j-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight which is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov’s formula where the spectral data is re...