arXiv subjects

Thomas Oliver

Publications and source records attributed to Thomas Oliver.

23 records · Page 2Linked to original sources

arXiv2020

Twisting moduli for GL(2)

We prove various converse theorems for automorphic forms on Γ_0(N), each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N is 18, 20, or 24 - these integers are the smallest multiples of 4 or 9 not covered by earlier work of Conrey-Farmer. This development is a consequence of finding generating sets for Γ_0(N) such that each generator can be written as a product of special matrices. As for real-analytic Maass forms of even (resp. odd) weight we prove the analogous statement for N=1,...12,16,18 (resp. N=1,...,12,14,15,16,17,18,20,23,24).

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arXiv2017

A conjectural extension of Hecke's converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.

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arXiv2016

Notes on Low Degree L-Data

These notes are an extended version of a talk given by the author at the conference "Analytic Number Theory and Related Areas", held at Research Institute for Mathematical Sciences, Kyoto University in November 2015. We are interested in "$L$-data", an axiomatic framework for $L$-functions introduced by Andrew Booker in 2013. Associated to each $L$-datum, one has a real number invariant known as the degree. Conjecturally the degree $d$ is an integer. Moreover, if $d\in\mathbb{N}$ then one expects that the $L$-datum is that of a $GL_n(\mathbb{A}_F)$-automorphic representation, for some number field $F$. In fact, if $F=\mathbb{Q}$, then $n=d$. This statement was shown to be true for $0\leq d<5/3$ by Booker in his pioneering paper, and in these notes we consider an extension of his methods to $0\leq d<2$. This is simultaneously a generalisation of Booker's result and the results and techniques of Kaczorowski--Perelli in the Selberg class (the best known to date). Furthermore, we consider applications to zeros of automorphic $L$-functions. In these notes we review Booker's results and announce new ones to appear elsewhere shortly.

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arXiv2015

Automorphicity and Mean-Periodicity

If C is a smooth projective curve over a number field k, then, under fair hypotheses, its L-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is X-mean-periodic for some appropriate functional space X. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such L-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of GL_{2g}-automorphic representations to certain vector spaces which are constructed from the Mellin transforms of specified products of arithmetic zeta functions corresponding to a model of the curve and the base number field. To compare automorphicity and mean-periodicity, we use a technique emulating the Rankin-Selberg method, in which the function mean-periodic function plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic $L$-functions.

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arXiv2015

Zeta Functions on Arithmetic Surfaces

We use a form of lifted harmonic analysis to develop a two-dimensional adelic integral representation of the zeta functions of simple arithmetic surfaces. Manipulations of this integral then lead to an adelic interpretation of the so-called mean-periodicity correspondence, which is comparable to the better known automorphicity conjectures for the generic fibre.

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