arXiv subjects

George J. Schaeffer

Publications and source records attributed to George J. Schaeffer.

3 recordsLinked to original sources

arXiv2021

Totally real algebraic integers of arboreal height 2

In arXiv:1302.4423, Salez proved that every totally real algebraic integer is the eigenvalue of some tree. We define the "arboreal height" of a totally real algebraic integer $λ$ to be the minimal height of a rooted tree having $λ$ as an eigenvalue. In this paper, we prove several results about totally real algebraic integers of arboreal height $2$: We show that all real quadratic integers have arboreal height $\le 2$. We characterize the totally real cubic integers of arboreal height $2$. Finally, we prove that every totally irrational real number field is generated (as a ring over $\mathbb{Q}$) by some $λ$ of arboreal height $2$.

math.NT↗

arXiv2016

Detecting large simple rational Hecke modules for $Γ_0(N)$ via congruences

We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_2(Γ_0(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_0(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. For prime levels $N\equiv 7\mod 8$ our method yields an unconditional bound of $d\ge\log_2\log_2(N/8)$, improving the known bound of $d\gg\sqrt{\log\log N}$ due to Murty--Sinha and Royer. We also discuss conditional bounds, the strongest of which is $d\gg_εN^{1/2-ε}$ over a large set of primes $N$, contingent on Soundararajan's heuristics for the class number problem and Artin's conjecture on primitive roots. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_k(\mathrm{SL}_2(\mathbb{Z});\mathbb{Q})$.

math.NT↗

arXiv2014

Hecke stability and weight 1 modular forms

The Galois representations associated to weight $1$ newforms over $\bar{\mathbb{F}}_p$ are remarkable in that they are unramified at $p$, but the computation of weight $1$ modular forms has proven to be difficult. One complication in this setting is that a weight $1$ cusp form over $\bar{\mathbb{F}}_p$ need not arise from reducing a weight $1$ cusp form over $\bar{\mathbb{Q}}$. In this article we propose a unified "Hecke stability method" for computing spaces of weight $1$ modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a finite-dimensional Hecke module of ratios of modular forms must consist of genuine modular forms. We conclude with some applications of the Hecke stability method motivated by the refined inverse Galois problem.

math.NT↗