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Rachel Newton

Publications and source records attributed to Rachel Newton.

At least 19 recordsLinked to original sources
arXiv2026

The Hasse norm principle for $A_4$-quartic extensions of global function fields

For a finite extension of global fields $K/k$, the norm map $N_{K/k} : K^\times \to k^\times$ extends to a map on id\`ele groups. The Hasse norm principle holds if every element of $k^\times$ that is a norm everywhere locally is also a norm globally. In this paper, we study the statistics of the Hasse norm principle in a setting that is out of reach in the number field context, namely that of $A_4$-quartic extensions. We show that failures of the Hasse norm principle are generally rare for $A_4$-quartic extensions of global function fields $\mathbb{F}_q(t)$. We achieve this by introducing a decorated Hurwitz space parametrising the failures of the Hasse norm principle and then using the Chebotarev density theorem to estimate their frequency.

math.NT↗
arXiv2026

Refined obstructions to local-global principles for 0-cycles

We introduce new `refined' obstructions to local-global principles for 0-cycles on algebraic varieties over number fields. Assuming finiteness of relevant Tate--Shafarevich groups, we show that the Hasse principle and weak approximation for 0-cycles on generalised Kummer varieties and bielliptic surfaces are controlled by obstructions of this new type. As an additional application of our refined obstructions, we answer a question of Zhang about the relationship between the Brauer--Manin and connected descent obstructions for 0-cycles. We also show that a Corwin--Schlank style refined obstruction set coincides with the set of global 0-cycles, conditionally on the Section Conjecture.

math.AG↗
arXiv2025

Wild Brauer classes via prismatic cohomology

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $X$ a smooth proper $K$-variety with good reduction. Under a mild assumption on the behaviour of Hodge numbers under reduction modulo $p$, we prove that the existence of a non-zero global 2-form on $X$ implies, after a finite extension of $K$, the existence of $p$-torsion Brauer classes with surjective evaluation map. This implies that any smooth proper variety over a number field which satisfies weak approximation over all finite extensions has no non-zero global 2-form. The proof is based on a prismatic interpretation of Brauer classes with eventually constant evaluation, and a Newton-above-Hodge result for the mod $p$ reduction of prismatic cohomology. This generalises work of Bright and the second-named author beyond the ordinary reduction case.

math.AG↗
arXiv2024

Machine learning for modular multiplication

Motivated by cryptographic applications, we investigate two machine learning approaches to modular multiplication: namely circular regression and a sequence-to-sequence transformer model. The limited success of both methods demonstrated in our results gives evidence for the hardness of tasks involving modular multiplication upon which cryptosystems are based.

cs.LG↗
arXiv2023

Transcendental Brauer-Manin obstructions on singular K3 surfaces

Let E and E' be elliptic curves over Q with complex multiplication by the ring of integers of an imaginary quadratic field K and let Y=Kum(ExE') be the minimal desingularisation of the quotient of ExE' by the action of -1. We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer-Manin obstructions to weak approximation.

math.NT↗
arXiv2022

Distribution of genus numbers of abelian number fields

We study the quantitative behaviour of genus numbers of abelian extensions of number fields with given Galois group. We prove an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time.

math.NT↗
arXiv2022

Explicit uniform bounds for Brauer groups of singular K3 surfaces

Let $k$ be a number field. We give an explicit bound, depending only on $[k:\mathbf{Q}]$ and the discriminant of the Néron--Severi lattice, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on $[k:\mathbf{Q}]$ and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on $[k:\mathbf{Q}]$, on the number of $\mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.

math.NT↗
arXiv2021

Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensions

Let $K/k$ be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for $K/k$ and the defect of weak approximation for the norm one torus $R^1_{K/k} \mathbb{G}_m$. We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of $K/k$ has symmetric or alternating Galois group.

math.NT↗
arXiv2020

Evaluating the wild Brauer group

Classifying elements of the Brauer group of a variety X over a p-adic field according to the p-adic accuracy needed to evaluate them gives a filtration on Br X. We relate this filtration to that defined by Kato's Swan conductor. The refined Swan conductor controls how the evaluation maps vary on p-adic discs: this provides a geometric characterisation of the refined Swan conductor. We give applications to rational points on varieties over number fields, including failure of weak approximation for varieties admitting a non-zero global 2-form.

math.AG↗
arXiv2020

The Hasse Norm Principle in Global Function Fields

Let $L$ be a finite extension of $\mathbb{F}_q(t)$. We calculate the proportion of polynomials of degree $d$ in $\mathbb{F}_q[t]$ that are everywhere locally norms from $L/\mathbb{F}_q(t)$ which fail to be global norms from $L/\mathbb{F}_q(t)$.

math.NT↗
arXiv2019

A bound on the primes of bad reduction for CM curves of genus 3

We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields.

math.NT↗
arXiv2018

Arithmetic of rational points and zero-cycles on Kummer varieties

Let $k$ be a number field, let $X$ be a Kummer variety over $k$, and let $δ$ be an odd integer. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ for $X$. For example, we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on $X$ over all finite extensions of $k$, then the $2$-primary Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of degree $δ$ on $X$ over $k$.

math.NT↗
arXiv2018

Number fields with prescribed norms

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100\%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

math.NT↗
arXiv2018

Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces

Let $k$ be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ for Kummer varieties over $k$. For example, for any Kummer variety $X$ over $k$, we show that if the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rational points on $X$ over all finite extensions of $k$, then the ($2$-primary) Brauer-Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on $X$ over $k$. We also obtain similar results for products of Kummer varieties, K3 surfaces and rationally connected varieties.

math.NT↗
arXiv2017

Non-ordinary curves with a Prym variety of low $p$-rank

If $π: Y \to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_π$ is a principally polarized abelian variety of dimension $g-1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$, it is not known in general which $p$-ranks can occur for $P_π$ under restrictions on the $p$-rank of $X$. In this paper, when $X$ is a non-hyperelliptic curve of genus $g=3$, we analyze the relationship between the Hasse-Witt matrices of $X$ and $P_π$. As an application, when $p \equiv 5 \bmod 6$, we prove that there exists a curve $X$ of genus $3$ and $p$-rank $f=3$ having an unramified double cover $π:Y \to X$ for which $P_π$ has $p$-rank $0$ (and is thus supersingular); for $3 \leq p \leq 19$, we verify the same for each $0 \leq f \leq 3$. Using theoretical results about $p$-rank stratifications of moduli spaces, we prove, for small $p$ and arbitrary $g \geq 3$, that there exists an unramified double cover $π: Y \to X$ such that both $X$ and $P_π$ have small $p$-rank.

math.NT↗
arXiv2016

Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Q_p. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.

math.NT↗
arXiv2016

Orbifold autoequivalent exceptional unimodal singularities

We prove that different expressions of the same exceptional unimodal singularity are orbifold equivalent. As in our previous paper, the matrix factorizations proving these orbifold equivalences depend again on certain parameters satisfying some equations whose solutions are permuted by Galois groups.

math.QA↗