arXiv subjects

Alex Cowan

Publications and source records attributed to Alex Cowan.

13 recordsLinked to original sources

arXiv2025

The relative entropy of primes in arithmetic progressions is really small

Fix a modulus $q$. One would expect the number of primes in each invertible residue class mod $q$ to be multinomially distributed, i.e. for each $p \,\mathrm{mod}\, q$ to behave like an independent random variable uniform on $(\mathbb{Z}/q\mathbb{Z})^\times$. Using techniques from data science, we discover overwhelming evidence to the contrary: primes are much more uniformly distributed than iid uniform random variables. This phenomenon was previously unknown, and there is no clear theoretical explanation for it. To demonstrate that our test statistic of choice, the KL divergence, is indeed extreme, we prove new bounds for the left tail of the relative entropy of the uniform multinomial using the method of types.

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arXiv2025

On the mean value of $\mathrm{GL}_1$ and $\mathrm{GL}_2$ $L$-functions, with applications to murmurations

"Murmurations" are a recently-discovered type of fine structure in sums of Dirichlet coefficients averaged over families of $L$-functions. The root cause of this phenomenon remains mysterious. In the present paper, we demonstrate how murmurations arise from the averaging of approximate functional equations. This approach to the study of murmurations explains their empirically observed ubiquity, as well as their characteristic scale invariance and the peculiar normalization they demand. We implement our new approach to the study of murmurations in the case of quadratic twist families of $\mathrm{GL}_1$ automorphic representations, where we exhibit murmurations unconditionally. Our proof centres around estimating mean values of the $L$-functions in our quadratic twist families. In particular, we require estimates valid significantly higher in the critical strip than what existing results provide. To produce these estimates, we construct a variation of the approximate functional equation which is imbued with a mechanism for dynamically rebalancing error terms while preserving holomorphicity. We also generalize and sharpen results of Jutila and Stankus on sums of quadratic characters and fundamental discriminants. Mean value estimates are given for $\mathrm{GL}_2$ quadratic twist families as well.

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arXiv2024

Murmurations and ratios conjectures

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a fixed elliptic curve, and the inverse Mellin transform of the shifted second moment of $\zeta'/\zeta$ on vertical lines; assuming GRH, for primitive quadratic Dirichlet characters, and holomorphic modular forms of prime level tending to infinity with sign and weight fixed; and unconditionally the inverse Mellin transform of the shifted second moment of $\zeta$ on vertical lines. We also present a generalization of our approach which relies only on the approximate functional equation in place of ratios conjectures.

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arXiv2024

Conductor distributions of elliptic curves

We determine the distribution of the conductors $N$ of rational elliptic curves when ordered by naive height $H$, in the form of an explicit density function for the ratios $N/H$. Our work is essentially an effective version of the Brumer--McGuinness--Watkins heuristic. Applying our results to the problem of enumerating elliptic curves by conductor gives the strongest bounds yet for the number of elliptic curves which have conductor much smaller than their height for ranges up to $H \ll N^{1.2165}$.

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arXiv2024

Generic models for genus 2 curves with real multiplication

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental discriminants $D < 40$. A key step in our proof is to develop an algorithm for minimisation of conic bundles fibred over $\mathbb{P}^2$. We apply this algorithm to simplify the equations for the Mestre conic associated to the generic point on the Hilbert modular surface of fundamental discriminant $D < 100$ computed by Elkies--Kumar.

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arXiv2023

Murmurations and explicit formulas

Unexpected oscillations in $a_p$ values in a family of elliptic curves were observed experimentally by He, Lee, Oliver, and Pozdnyakov. We propose a heuristic explanation for these oscillations based on the "explicit formula" from analytic number theory. A crucial ingredient in this heuristic is that the distribution of the zeros of the associated $L$-functions has a quasi-periodic structure. We present empirical results for a family of elliptic curves, a family of quadratic Dirichlet characters whose values exhibit similar oscillations, and a family of Dirichlet characters whose values do not.

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arXiv2023

A twisted additive divisor problem

We give asymptotics for shifted convolutions of the form $$\sum_{n < X} \frac{\sigma_{2u}(n,\chi)\sigma_{2v}(n+k,\psi)}{n^{u+v}}$$ for nonzero complex numbers $u,v$ and nontrivial Dirichlet characters $\chi,\psi$. We use the technique of "automorphic regularization" to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.

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arXiv2023

Paired comparisons for games of chance

We present a Bayesian rating system based on the method of paired comparisons. Our system is a flexible generalization of the well-known Glicko, and in particular can better accommodate games with significant elements of luck. Our system is currently in use in the online game Duelyst II, and in that setting outperforms Glicko2.

stat.ME↗

arXiv2023

Counting modular forms by rationality field

We investigate the distribution of degrees and rationality fields of weight 2 newforms. In particular, we give heuristic upper bounds on how often degree $d$ rationality fields occur for squarefree levels, and predict finiteness if $d \ge 7$. When $d=2$, we make predictions about how frequently specific quadratic fields occur, prove lower bounds, and conjecture that $\mathbb{Q}(\sqrt 5)$ is the most common quadratic rationality field.

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arXiv2022

Moduli for rational genus 2 curves with real multiplication for discriminant 5

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their Jacobians, but the converse is not true. We give a simple generic description of which rational moduli points correspond to rational curves, as well as give associated Weierstrass models, in the case of RM by the ring of integers of $\mathbb{Q}(\sqrt{5})$. To prove this, we provide some techniques for reducing quadratic forms over polynomial rings.

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arXiv2020

Computing newforms using supersingular isogeny graphs

We describe an algorithm that we used to compute the q-expansions of all weight 2 cusp forms of prime level at most 2,000,000 and dimension at most 6. We also present an algorithm that we used to verify that there was only one cusp form of dimension 7 or more per Atkin-Lehner eigenspace for prime levels between 10,000 and 1,000,000. Our algorithm is based on Mestre's M\'ethode des Graphes, and involves supersingular isogeny graphs and Wiedemann's algorithm for finding the minimal polynomial of sparse matrices over finite fields.

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arXiv2020

Conjecture: 100% of elliptic surfaces over $\mathbb{Q}$ have rank zero

Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain $L$-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.

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arXiv2019

The distribution of multiples of real points on an elliptic curve

Given an elliptic curve $E$ and a point $P$ in $E(\mathbb{R})$, we investigate the distribution of the points $nP$ as $n$ varies over the integers, giving bounds on the $x$ and $y$ coordinates of $nP$ and determining the natural density of integers $n$ for which $nP$ lies in an arbitrary open subset of $\mathbb{R}^2$. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.

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