Andrew R. Booker

Effective computation of Maass cusp forms

Author please provide the abstract. Please provide the abstract of this paper that should not exceed 150 words (including spaces) and citation free. 1 Preliminary The aim of this paper is to address theoretical and practical aspects of high-precision computation of Maass forms. Namely, we compute to over 1000 decimal places the Laplacian and Hecke eigenvalues for the first few Maass forms on PSL(2, Z)\H, and certify the Laplacian eigenvalues correct to 100 places. We then use these computations to test certain algebraicity properties of the coefficients. The outline of the paper is as follows. In Section 2, we discuss Hejhal’s algorithm for computation of Maass forms on cofinite Fuchsian groups with cusps, and the details necessary to implement it in high precision. This algorithm is heuristic and does not prove the existence of cusp forms. In Section 3 we turn to the question of rigorously verifying that a proposed eigenvalue, together with a proposed set of Fourier coefficients, indeed correspond to a true Maass cusp form. We will use standard methods to show that the putative eigenfunction has almost all of its spectral support concentrated near the proposed eigenvalue. It is a more subtle point to show that it is close to a cusp form

Numerical computations with the trace formula and the Selberg eigenvalue conjecture

Abstract We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary and analytic theory of numbers, Banach Center Publ. 17, Warsaw (1985), 217–306.]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1/4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois representations of small squarefree conductor.

A database of genus-2 curves over the rational numbers

We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. This data has been incorporated into the L-Functions and Modular Forms Database (LMFDB).

Bounds and algorithms for the K-Bessel function of imaginary order

Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function

Zeros of L-functions outside the critical strip

{K}_{ir} (x)

Simple zeros of degree 2 L-functions

of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of

Detecting squarefree numbers

{K}_{ir} (x)

L-functions as distributions

and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of

Primitive values of quadratic polynomials in a finite field

r

Turing's method for the Selberg zeta-function

. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of