arXiv subjects

Sam Frengley

Publications and source records attributed to Sam Frengley.

10 recordsLinked to original sources

arXiv2026

Tschirnhausen bundles of sextic covers of $\mathbb{P}^1$

A degree $d$ genus $g$ cover of the complex projective line by a smooth irreducible curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. We classify the bundles that arise this way when $d = 6$. Interestingly, our methods show that all constraints on the pushforward are ``explained'' by multiplication in an algebra. Finally, we show that all possible pushforwards are realized by covers with a nontrivial proper subcover.

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arXiv2026

Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups

We exhibit examples of geometrically simple abelian surfaces $A/\mathbb{Q}$ with conductor bounded by $(10\,000)^2$ whose Tate--Shafarevich groups contain a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^2$ for each $p = 5, 7, 11, 13$. To find these examples we generalise work of Cremona--Freitas to give a candidate list of all congruences of a certain type between pairs of weight $2$ newforms $f \in S_2^{\mathrm{new}}(\Gamma_0(N))$ and $g \in S_2^{\mathrm{new}}(\Gamma_0(M))$ contained in the LMFDB (i.e., with $N, M \leq 10\,000$) and with coefficient fields of degree $\leq 4$. Passing from the modular forms to the corresponding abelian varieties we use visibility to (unconditionally) prove the existence of non-trivial elements of the Tate--Shafarevich group. Finally we construct an example of an abelian surface with $(\mathbb{Z}/7\mathbb{Z})^2 \subset \mathrm{Sha}(A/\mathbb{Q})$ which is (conjecturally) not visible in any abelian threefold.

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arXiv2025

Tschirnhausen Bundles of Quintic Covers of $\mathbb{P}^1$

A degree $d$ genus $g$ cover of the complex projective line by a smooth irreducible curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. We classify the bundles that arise this way when $d = 5$. Equivalently, we classify which $\mathbb{P}^3$-bundles over $\mathbb{P}^1$ contain smooth irreducible degree $5$ covers of $\mathbb{P}^1$. Our main contribution is proving the existence of \emph{smooth} covers whose structure sheaf has the desired pushforward. We do this by showing that the substack of singular curves has positive codimension in the moduli stack of finite flat covers with desired pushforward. To compute the dimension of the space of singular curves, we prove a (relative) ``minimization theorem'', which is the geometric analogue of Bhargava's sieving argument when computing the densities of discriminants of quintic number fields.

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arXiv2025

Galois groups of simple abelian varieties over finite fields and exceptional Tate classes

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate conjecture in cases when the angle rank is non-maximal. Our primary tool is a precise combinatorial condition which, given a geometrically simple abelian variety $A/\mathbf{F}_q$ with commutative endomorphism algebra, describes whether $A$ has exceptional classes (i.e., $\mathrm{Gal}( \bar{\mathbf{F}}_q/\mathbf{F}_q)$-invariant classes in $H_{\text{\'et}}^{2r}(A_{\bar{\mathbf{F}}_q}, \mathbf{Q}_\ell(r))$ not contained in the span of classes of intersections of divisors). The criterion depends only on the Galois group of the minimal polynomial of Frobenius and its action on the Newton polygon of $A$. Our tools provide substantial control over the isogeny invariants of $A$, allowing us to prove a number of new results. Firstly, we provide an algorithm which, given a Newton polygon and CM field, determines if they arise from a geometrically simple abelian variety $A/\mathbf{F}_q$ and, if so, outputs one such $A$. As a consequence we show that every CM field occurs as the center of the endomorphism algebra of an abelian variety $A/\mathbf{F}_q$. Secondly, we refine a result of Tankeev and Dupuy--Kedlaya--Zureick-Brown on angle ranks of abelian varieties. In particular, we show that ordinary geometrically simple varieties of prime dimension have maximal angle rank.

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arXiv2024

Galois groups of low dimensional abelian varieties over finite fields

We consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the weighted permutation representation which encompasses all three of these invariants and use it to study the subtle relationships between them. We use this permutation representation to classify the triples of invariants that occur for abelian surfaces and simple abelian threefolds.

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arXiv2024

On the geometry of the Humbert surface of square discriminant

For every positive integer $N$ we determine the Enriques--Kodaira type of the Humbert surface of discriminant $N^2$ which parametrises principally polarised abelian surfaces that are $(N,N)$-isogenous to a product of elliptic curves. A key step in the proof is to analyse the fixed point locus of a Fricke-like involution on the Hilbert modular surface of discriminant $N^2$ which was studied by Hermann and by Kani and Schanz. To this end, we construct certain "diagonal" Hirzebruch--Zagier divisors which are fixed by this involution. In our analysis we obtain a genus formula for these divisors, which includes the case of modular curves associated to (any) extended Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ and which may be of independent interest.

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arXiv2024

Explicit $7$-torsion in the Tate-Shafarevich groups of genus $2$ Jacobians

Let $C/\mathbb{Q}$ be a genus $2$ curve whose Jacobian $J/\mathbb{Q}$ has real multiplication by a quadratic order in which $7$ splits. We describe an algorithm which outputs twists of the Klein quartic curve which parametrise elliptic curves whose mod $7$ Galois representations are isomorphic to a sub-representation of the mod $7$ Galois representation attached to $J/\mathbb{Q}$. Applying this algorithm to genus $2$ curves of small conductor in families of Bending and Elkies--Kumar we exhibit a number of genus $2$ Jacobians whose Tate--Shafarevich groups (unconditionally) contain a non-trivial element of order $7$ which is visible in an abelian three-fold.

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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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arXiv2022

On $12$-congruences of elliptic curves

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over $\mathbb{Q}$ with $12$-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of $12$-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the $2$, $3$, or $4$-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the $j$-invariants of the pair of elliptic curves.

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arXiv2021

Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$. By computing explicit models for these double covers we find all pairs $(N, r)$ such that there exist infinitely many $j$-invariants of elliptic curves $E/\mathbb{Q}$ which are $N$-congruent with power $r$ to a quadratic twist of $E$. We also find an example of a $48$-congruence over $\mathbb{Q}$. We make a conjecture classifying nontrivial $(N,r)$-congruences between quadratic twists of elliptic curves over $\mathbb{Q}$. Finally, we give a more detailed analysis of the level $15$ case. We use elliptic Chabauty to determine the rational points on a modular curve of genus $2$ whose Jacobian has rank $2$ and which arises as a double cover of the modular curve $X(\mathrm{ns} 3^+, \mathrm{ns} 5^+)$. As a consequence we obtain a new proof of the class number $1$ problem.

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