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Adam Morgan

Publications and source records attributed to Adam Morgan.

At least 19 recordsLinked to original sources
arXiv2025

Constructing Jacobians of rank 1

Let $K$ be a number field, let $g \geq 1$ be an integer and let $f(x) = (x - a_1) \cdots (x - a_{2g + 1}) \in O_K[x]$ be a polynomial that splits into $2g + 1$ distinct linear factors. Write $C$ for the hyperelliptic curve given by $C: y^2 = f(x)$ and write $J = \mathrm{Jac}(C)$ for its Jacobian. Under mild technical assumptions on $f$ that are satisfied almost always, we prove that there exists some $d \in K^\times$ such that the quadratic twist $J^d$ has rank exactly equal to $1$. As a consequence, we deduce that for any positive integer $g$, there exists an absolutely simple abelian variety over $K$ with dimension equal to $g$ and rank equal to $1$.

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arXiv2025

Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups

We address several seemingly disparate problems in arithmetic geometry: the statistical behaviour of the Galois module structure of Mordell--Weil groups of a fixed elliptic curve over varying quadratic extensions; the frequency of failure of the Hasse principle in quadratic twist families of genus $1$ hyperelliptic curves; and the Hasse principle for Kummer varieties. The common technical ingredient for all of these is a result on the distribution of $2$-Selmer ranks in certain sparse families of quadratic twists of a given abelian variety.

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arXiv2024

A note on local formulae for the parity of Selmer ranks

In this note, we provide evidence for a certain twisted version of the parity conjecture for Jacobians, introduced in prior work of V. Dokchitser, Green, Konstantinou and the author. To do this, we use arithmetic duality theorems for abelian varieties to study the determinant of certain endomorphisms acting on p-infinity Selmer groups.

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arXiv2024

Hasse principle for intersections of two quadrics via Kummer surfaces

We prove new cases of the Hasse principle for Kummer surfaces constructed from 2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant Tate-Shafarevich groups. Under the same assumption, we deduce the Hasse principle for quartic del Pezzo surfaces with trivial Brauer group and irreducible or completely split characteristic polynomial, hence the Hasse principle for smooth complete intersections of two quadrics in the projective space of dimension at least 5.

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arXiv2024

On Galois covers of curves and arithmetic of Jacobians

We study the arithmetic of curves and Jacobians endowed with the action of a finite group $G$. This includes a study of the basic properties, as $G$-modules, of their $\ell$-adic representations, Selmer groups, rational points and Shafarevich-Tate groups. In particular, we show that $p^\infty$-Selmer groups are self-dual $G$-modules, and give various `$G$-descent' results for Selmer groups and rational points. Along the way we revisit, and slightly refine, a construction going back to Kani and Rosen for associating isogenies to homomorphisms between permutation representations. With a view to future applications, it is convenient to work throughout with curves that are not assumed to be geometrically connected (or even connected); such curves arise naturally when taking Galois closures of covers of curves. For lack of a suitable reference, we carefully detail how to deduce the relevant properties of such curves and their Jacobians from the more standard geometrically connected case.

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arXiv2023

Hasse principle for Kummer varieties in the case of generic 2-torsion

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov used Swinnerton-Dyer's descent-fibration method to establish the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety under certain largeness assumptions on its mod 2 Galois image. Their method breaks down however when the Galois image is maximal, due to the possible failure of the Shafarevich--Tate group of quadratic twists of A to have square order. In this work we overcome this obstruction by combining second descent ideas in the spirit of Harpaz and Smith with results on the parity of 2-infinity Selmer ranks in quadratic twist families. This allows Swinnerton-Dyer's method to be successfully applied to K3 surfaces arising as quotients of 2-coverings of Jacobians of genus 2 curves with no rational Weierstrass points.

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arXiv2022

Parity of ranks of Jacobians of curves

We investigate Selmer groups of Jacobians of curves that admit an action of a non-trivial group of automorphisms, and give applications to the study of the parity of Selmer ranks. Under the Shafarevich--Tate conjecture, we give an expression for the parity of the Mordell--Weil rank of an arbitrary Jacobian in terms of purely local invariants; the latter can be seen as an arithmetic analogue of local root numbers, which, under the Birch--Swinnerton-Dyer conjecture, similarly control parities of ranks of abelian varieties. As an application, we give a new proof of the parity conjecture for elliptic curves. The core of the paper is devoted to developing the arithmetic theory of Jacobians for Galois covers of curves, including decomposition of their L-functions, and the interplay between Brauer relations and Selmer groups.

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arXiv2022

Field change for the Cassels-Tate pairing and applications to class groups

In previous work, the authors defined a category $SMod_F$ of finite Galois modules decorated with local conditions for each global field $F$. In this paper, given an extension $K/F$ of global fields, we define a restriction of scalars functor from $SMod_K$ to $SMod_F$ and show that it behaves well with respect to the Cassels-Tate pairing. We apply this work to study the class groups of global fields in the context of the Cohen-Lenstra heuristics.

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arXiv2021

The Cassels-Tate pairing for finite Galois modules

Given a global field $F$ with absolute Galois group $G_F$, we define a category $SMod_F$ whose objects are finite $G_F$-modules decorated with local conditions. We define this category so that `taking the Selmer group' defines a functor $Sel$ from $SMod_F$ to $Ab$. After defining a duality functor $\vee$ on $SMod_F$, we show that every short exact sequence $0 \to M_1 \to M \to M_2 \to 0$ in $SMod_F$ gives rise to a natural bilinear pairing $$Sel (M_2) \times Sel (M_1^{\vee}) \to \mathbb{Q}/\mathbb{Z}$$ whose left and right kernels are the images of $Sel (M)$ and $Sel (M^{\vee})$, respectively. This generalizes the Cassels--Tate pairing defined on the Shafarevich--Tate group of an abelian variety over $F$ and results in a flexible theory in which pairings associated to different exact sequences can be readily compared to one another. As an application, we give a new proof of Poonen and Stoll's results concerning the failure of the Cassels--Tate pairing to be alternating for principally polarized abelian varieties and extend this work to the setting of Bloch--Kato Selmer groups.

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arXiv2021

The $4$-rank of class groups of $K(\sqrt{n})$

Let $K/\mathbb{Q}$ be a quadratic extension. In this paper we study the $4$-rank of the class group $\text{Cl}(K(\sqrt{n}))$, where $n$ varies over squarefree rational integers. We show that for $100\%$ of squarefree $n$, the $4$-rank is given by an explicit formula involving the $2$-rank of $\text{Cl}(K)$ and the number of prime factors of $n$ which are inert in $K/\mathbb{Q}$.

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arXiv2020

On 2-Selmer groups of twists after quadratic extension

Let $E/\mathbb{Q}$ be an elliptic curve with full rational 2-torsion. As d varies over squarefree integers, we study the behaviour of the quadratic twists $E_d$ over a fixed quadratic extension $K/\mathbb{Q}$. We prove that for 100% of twists the dimension of the 2-Selmer group over K is given by an explicit local formula, and use this to show that this dimension follows an Erd\H{o}s--Kac type distribution. This is in stark contrast to the distribution of the dimension of the corresponding 2-Selmer groups over $\mathbb{Q}$, and this discrepancy allows us to determine the distribution of the 2-torsion in the Shafarevich--Tate groups of the $E_d$ over K also. As a consequence of our methods we prove that, for 100% of twists d, the action of $\operatorname{Gal}(K/\mathbb{Q})$ on the 2-Selmer group of $E_d$ over K is trivial, and the Mordell--Weil group $E_d(K)$ splits integrally as a direct sum of its invariants and anti-invariants. On the other hand, we give examples of thin families of quadratic twists in which a positive proportion of the 2-Selmer groups over K have non-trivial $\operatorname{Gal}(K/\mathbb{Q})$-action, illustrating that the previous results are genuinely statistical phenomena.

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arXiv2020

A user's guide to the local arithmetic of hyperelliptic curves

A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.

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arXiv2018

Tate module and bad reduction

Let C/K be a curve over a local field. We study the natural semilinear action of Galois on the minimal regular model of C over a field F where it becomes semistable. This allows us to describe the Galois action on the l-adic Tate module of the Jacobian of C/K in terms of the special fibre of this model over F.

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arXiv2018

Arithmetic of hyperelliptic curves over local fields

We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distances between the roots of f(x), and show that this elementary combinatorial object encodes the curve's Galois representation, conductor, whether the curve is semistable, and if so, the special fibre of its minimal regular model, the discriminant of its minimal Weierstrass equation and other invariants.

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arXiv2017

Quadratic twists of abelian varieties and disparity in Selmer ranks

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree and we describe both.

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arXiv2017

Semistable types of hyperelliptic curves

In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining equation (`cluster pictures'). We construct explicit combinatorial one-to-one correspondences between the three, which furthermore respect automorphisms and allow to keep track of the monodromy pairing and the Tamagawa group of the Jacobian. We introduce a classification scheme and a naming convention for semistable types of hyperelliptic curves and types with a Frobenius action. This is the higher genus analogue of the distinction between good, split and non-split multiplicative reduction for elliptic curves. Our motivation is to understand $L$-factors, Galois representations, conductors, Tamagawa numbers and other local invariants of hyperelliptic curves and their Jacobians.

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arXiv2016

Cuntz-Pimsner Algebras and Twisted Tensor Products

Given two correspondences $X$ and $Y$ and a discrete group $G$ which acts on $X$ and coacts on $Y$, one can define a twisted tensor product $X\boxtimes Y$ which simultaneously generalizes ordinary tensor products and crossed products by group actions and coactions. We show that, under suitable conditions, the Cuntz-Pimsner algebra of this product, $\mathcal O_{X\boxtimes Y}$, is isomorphic to a "balanced" twisted tensor product $\mathcal O_X\boxtimes_\mathbb T\mathcal O_Y$ of the Cuntz-Pimsner algebras of the original correspondences. We interpret this result in several contexts and connect it to existing results on Cuntz-Pimsner algebras of crossed products and tensor products.

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