arXiv subjects

Andrew V. Sutherland

Publications and source records attributed to Andrew V. Sutherland.

At least 19 recordsLinked to original sources

arXiv2026

Rational torsion on simple genus two Jacobians

We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over $\mathbb Q$. The largest group, which has order 96 and invariants [2,2,2,12], is realized by curves of the form $y^2 = x(x-a^2)(x-b^2)(x-c^2)(x-u^2)(x-v^2)$ where $a,b,c,u,v$ are positive integers that satisfy $a^2 + b^2 + c^2 = u^2 + v^2$ and $a^4 + b^4 + c^4 = u^4 + v^4$. We also find realizations of the groups [2,2,20], [2,2,4,4], [2,2,2,8], [2,4,8], and [6,6]. Finally, we record, to the best of our knowledge, all known subgroups that arise in genus-two Jacobians over $\mathbb Q$, in the geometrically simple case and in general.

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arXiv2026

Abelian surfaces of small conductor from genus 3 double covers

We describe the construction of a database of roughly half a million abelian surfaces over Q of small conductor arising as Pryms associated to a genus 3 double cover of a genus 1 curve. Our construction uses a method that provides a degree of control over the primes of bad reduction.

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arXiv2025

Murmurations for elliptic curves ordered by height

He, Lee, Oliver, and Pozdnyakov~\cite{HLOP} have empirically observed that the average of the $p$th coefficients of the $L$-functions of elliptic curves of particular ranks in a given range of conductors $N$ appears to approximate a continuous function of $p$, depending primarily on the parity of the rank. Hence the sum of $p$th coefficients against the root number also appears to approximate a continuous function, dubbed the murmuration density. However, it is not clear from this numerical data how to obtain an explicit formula for the murmuration density. Convergence of similar averages was proved by Zubrilina~\cite{Zubrilina} for modular forms of weight $2$ (of which elliptic curves form a thin subset) and analogous results for other families of automorphic forms have been obtained in further work~\cite{BBLLD,LOP}. Each of these works gives an explicit formula for the murmuration density. We consider a variant problem where the elliptic curves are ordered by naive height, and the $p$th coefficients are averaged over $p/N$ in a fixed interval. We give a conjecture for the murmuration density in this case, as an explicit but complicated sum of Bessel functions. This conjecture is motivated by a theorem about a variant problem where we sum the $n$th coefficients for $n$ with no small prime factors against a smooth weight function. We test this conjecture for elliptic curves of naive height up to $2^{28}$ and find good agreement with the data. The theorem is proved using the Voronoi summation formula, and the method should apply to many different families of $L$-functions. By a similar approach, we give a prediction murmuration density for elliptic curves of prime conductor, ordered by conductor, again matching the data but lacking a motivating theorem. This is the first work to give an explicit formula for the murmuration density of a family of elliptic curves, in any ordering.

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arXiv2025

Decomposing a factorial into large factors

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erd\H{o}s and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible.

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arXiv2025

Generative Modeling for Mathematical Discovery

We present a new implementation of the LLM-driven genetic algorithm {\it funsearch}, whose aim is to generate examples of interest to mathematicians and which has already had some success in problems in extremal combinatorics. Our implementation is designed to be useful in practice for working mathematicians; it does not require expertise in machine learning or access to high-performance computing resources. Applying {\it funsearch} to a new problem involves modifying a small segment of Python code and selecting a large language model (LLM) from one of many third-party providers. We benchmarked our implementation on three different problems, obtaining metrics that may inform applications of {\it funsearch} to new problems. Our results demonstrate that {\it funsearch} successfully learns in a variety of combinatorial and number-theoretic settings, and in some contexts learns principles that generalize beyond the problem originally trained on.

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arXiv2025

Completely decomposable modular Jacobians

We use recently developed algorithms and a new database of modular curves constructed for the L-functions and Modular Forms Database to enumerate completely decomposable modular Jacobians of level N < 240. In particular, we find examples in 13 previously unknown genera of Jacobian varieties isogenous to a product of elliptic curves over Q. The new genera are: 38, 68, 75, 76, 77, 78, 113, 135, 137, 157, 159, 169, and 409.

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arXiv2025

Counting points on superelliptic curves in average polynomial time

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\ge 2$ and $f\in \mathbb Z[x]$ any squarefree polynomial of degree $d\ge 3$, along with a positive integer $N$. It can compute $\#X(\mathbb F_p)$ for all $p\le N$ not dividing $m\mathrm{lc}(f)\mathrm{disc}(f)$ in time $O(md^3 N\log^3 N\log\log N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$.

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arXiv2025

Computing Euler factors of genus 2 curves at odd primes of almost good reduction

We present an efficient algorithm to compute the Euler factor of a genus 2 curve C/Q at an odd prime p that is of bad reduction for C but of good reduction for the Jacobian of C (a prime of ``almost good'' reduction). Our approach is based on the theory of cluster pictures introduced by Dokchitser, Dokchitser, Maistret, and Morgan, which allows us to reduce the problem to a short, explicit computation over Z and Fp, followed by a point-counting computation on two elliptic curves over Fp, or a single elliptic curve over Fp^2. A key feature of our approach is that we avoid the need to compute a regular model for C. This allows us to efficiently compute many examples that are infeasible to handle using the algorithms currently available in computer algebra systems such as Magma and Pari/GP.

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arXiv2025

$\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$

We discuss the $\ell$-adic case of Mazur's "Program B" over $\mathbb{Q}$, the problem of classifying the possible images of $\ell$-adic Galois representations attached to elliptic curves $E$ over $\mathbb{Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell=2$ and $\ell\ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$. For each of these $\ell$, we compute the directed graph of arithmetically maximal $\ell$-power level modular curves $X_H$, compute explicit equations for all but three of them, and classify the rational points on all of them except $X_{\rm ns}^{+}(N)$, for $N = 27, 25, 49, 121$, and two level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$. Aside from the $\ell$-adic images that are known to arise for infinitely many $\bar{\mathbb{Q}}$-isomorphism classes of elliptic curves $E/\mathbb{Q}$, we find only 22 exceptional images that arise for any prime $\ell$ and any $E/\mathbb{Q}$ without complex multiplication; these exceptional images are realized by 20 non-CM rational $j$-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\rm ns}^+(\ell)$ with $\ell\ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell$-adic images of Galois for any elliptic curve over $\mathbb{Q}$. In an appendix with John Voight we generalize Ribet's observation that simple abelian varieties attached to newforms on $Γ_1(N)$ are of ${\rm GL}_2$-type; this extends Kolyvagin's theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.

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arXiv2024

Doubly isogenous curves of genus two with a rational action of $D_6$

Let $C$ and $C'$ be curves over a finite field $K$, provided with embeddings $\iota$ and $\iota'$ into their Jacobian varieties. Let $D\to C$ and $D'\to C'$ be the pullbacks (via these embeddings) of the multiplication-by-$2$ maps on the Jacobians. We say that $(C,\iota)$ and $(C',\iota')$ are \emph{doubly isogenous} if $\mathrm{Jac}(C)$ and $\mathrm{Jac}(C')$ are isogenous over $K$ and $\mathrm{Jac}(D)$ and $\mathrm{Jac}(D')$ are isogenous over~$K$. When we restrict attention to the case where $C$ and $C'$ are curves of genus $2$ whose groups of $K$-rational automorphisms are isomorphic to the dihedral group $D_6$ of order $12$, we find many more doubly isogenous pairs than one would expect from reasonable heuristics. Our analysis of this overabundance of doubly isogenous curves over finite fields leads to the construction of a pair of doubly isogenous curves over a number field. That such a global example exists seems extremely surprising. We show that the Zilber--Pink conjecture implies that there can only be finitely many such examples. When we exclude reductions of this pair of global curves in our counts, we find that the data for the remaining curves is consistent with our original heuristic. Computationally, we find that doubly isogenous curves in our family of $D_6$ curves can be distinguished from one another by considering the isogeny classes of the Prym varieties of certain unramified covers of exponent $3$ and $4$. We discuss how our family of curves can be potentially be used to obtain a deterministic polynomial-time algorithm to factor univariate polynomials over finite fields via an argument of Kayal and Poonen.

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arXiv2023

Computing the endomorphism ring of an elliptic curve over a number field

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. More generally, given an elliptic curve E over a number field, one can use them to compute the endomorphism ring of E. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches.

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arXiv2023

Sato-Tate groups of abelian threefolds

Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura's theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in Gap, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.

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arXiv2022

Counting points on smooth plane quartics

We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a prime of good reduction. We consider two approaches for computing $\#X(\mathbb F_p)$, one which runs in $O(p\log p\log\log p)$ time using $O(\log p)$ space and one which runs in $O(p^{1/2}\log^2\!p)$ time using $O(p^{1/2}\log p)$ space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $X/\mathbb Q$ that compute $\#X(\mathbb F_p)$ for good primes $p\le N$ in $O(N\log^3\! N)$ time using $O(N)$ space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $\mathbb P^1$, which in combination with previous results addresses all curves of genus $g\le 3$. Our algorithms also compute Cartier-Manin/Hasse-Witt matrices that may be of independent interest.

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arXiv2022

Computing classical modular forms

We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and Modular Forms Database (LMFDB).

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arXiv2022

Computing L-series of hyperelliptic curves

We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.

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arXiv2022

Computing images of Galois representations attached to elliptic curves

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasi-linear in n. We have applied our algorithms to the non-CM elliptic curves in Cremona's tables and the Stein--Watkins database, some 140 million curves of conductor up to 10^10, thereby obtaining a conjecturally complete list of 63 exceptional Galois images G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.

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arXiv2021

Sato-Tate Distributions

In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g <= 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry.

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arXiv2021

Stronger arithmetic equivalence

Motivated by a recent result of Prasad, we consider three stronger notions of arithmetic equivalence: local integral equivalence, integral equivalence, and solvable equivalence. In addition to having the same Dedekind zeta function (the usual notion of arithmetic equivalence), number fields that are equivalent in any of these stronger senses must have the same class number, and solvable equivalence forces an isomorphism of adele rings. Until recently the only nontrivial example of integral and solvable equivalence arose from a group-theoretic construction of Scott that was exploited by Prasad. Here we provide infinitely many distinct examples of solvable equivalence, including a family that contains Scott's construction as well as an explicit example of degree 96. We also construct examples that address questions of Scott, and of Guralnick and Weiss, and shed some light on a question of Prasad.

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