arXiv subjects

Pip Goodman

Publications and source records attributed to Pip Goodman.

6 recordsLinked to original sources

arXiv2025

Semistable abelian varieties over $\mathbb{Q}$ with bad reduction at 19 only: an overview of the Fontaine--Schoof strategy

In this paper we provide an overview of a strategy pioneered by Fontaine and heavily refined by Schoof to classify abelian varieties with prescribed bad reduction. Throughout the overview, we prove various non-trivial background results turning it into an introduction for readers unacquainted with this topic. With the overview completed, we provide explicit examples of the strategy in action. At first we give introductory examples, classifying semistable abelian varieties over $\mathbb{Q}$ with bad reduction at exactly one of 3 or 5 up to isogeny over $\mathbb{Q}$. We then move onto a harder example, proving the analogous result for 19, which is new.

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arXiv2025

Abelian threefolds with imaginary multiplication

Let $A$ be an abelian threefold defined over a number field $K$ with potential multiplication by an imaginary quadratic field $M$. Under mild assumptions on $K$, if $A$ has signature $(2,1)$ and the multiplication by $M$ is defined over $KM$, we attach to $A$ an elliptic curve defined over $K$ with potential complex multiplication by $M$, whose attached Galois representation is determined by the Hecke character associated to the determinant of the compatible system of $\lambda$-adic representations of $A$. We deduce that if the geometric endomorphism algebra of $A$ is an imaginary quadratic field, then it necessarily has class number bounded by $[KM:M]$.

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arXiv2021

Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field

In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be defined over $K(A[2])$, the field generated by its 2-torsion. When $K= \mathbb{Q}$ and $\mathrm{Gal}(\mathbb{Q}(A[2])/\mathbb{Q})$ is cyclic of prime order $p = 2 \dim(A) +1$, we prove that there are only finitely many possibilities for the geometric endomorphism algebra $\mathrm{End}(A) \otimes \mathbb{Q}$.In fact, when $\dim (A) \not \in \{3,5,9,21,33,81\}$, we show $\mathrm{End}(A) \otimes \mathbb{Q}$ is a proper subfield of the $p$-th cyclotomic field. In particular, when $g=2$, $\mathrm{End}(A) \otimes \mathbb{Q}$ is isomorphic to either $\mathbb{Q}$ or $\mathbb{Q}(\sqrt{5})$.

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arXiv2021

Cubic and quartic points on modular curves using generalised symmetric Chabauty

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell--Weil group of its Jacobian.

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arXiv2020

Superelliptic curves with large Galois images

Let $r>2$ and $\ell$ be primes. In this paper we study the mod $\ell$ Galois representations attached to curves of the form $y^r = f(x)$ where $f$ is monic and has coefficients belonging to the $r$-th cyclotomic field. We provide conditions on the coefficients (and degree) of $f$ which allow one to verify the mod $\ell$ image is large outside of a (typically small) finite explicit set of primes. We allow all values of $r$ for which the $r$-th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of ${\rm GL}_2$-type which allows the ground field to have unramified extensions. In proving the large image result we give a classification of the maximal subgroups containing transvections of certain classical groups and describe (in many cases) the images of inertia groups. The exact mod $\ell$ image is governed by the "endomorphism character", a certain algebraic Hecke character which generalises the CM character. When $r=3$, we depict the image in its entirety. To the author's knowledge, this is the first accurate description in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included.

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arXiv2019

Restrictions on endomorphism rings of jacobians and their minimal fields of definition

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians $J$ for which the Galois group associated to their 2-torsion is insoluble and 'large' (relative to the dimension of $J$). In this paper we examine what happens when this Galois group merely contains an element of 'large' prime order. In doing so we obtain a partial converse to a result by Guralnick and Kedlaya on the endomorphism field.

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