arXiv subjects

Caleb Springer

Publications and source records attributed to Caleb Springer.

11 recordsLinked to original sources

arXiv2025

Isogeny graphs of abelian varieties and singular ideals in orders

Famously, Kohel proved that isogeny graphs of ordinary elliptic curves are beautifully structured objects, now called volcanos. We prove graph structural theorems for abelian varieties of any dimension with commutative endomorphism ring and containing a fixed locally Bass order, leveraging an ideal-theoretic perspective on isogeny graphs. This generalizes previous results, which relied on restrictive additional assumptions, such as maximal real multiplication, ordinary, and absolutely simple (Brooks, Jetchev, Wesolowski 2017). In particular, our work also applies to non-simple and non-ordinary isogeny classes. To obtain our results, we first prove a structure theorem for the lattice of inclusion of the overorders of a locally Bass order in an \'etale algebra which is of independent interest. This analysis builds on a careful study of local singularities of the orders. We include several examples of volcanoes and isogeny graphs exhibiting unexpected properties ultimately due to our more general setting.

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arXiv2024

First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability

In this paper, we study questions of definability and decidability for infinite algebraic extensions ${\bf K}$ of $\mathbb{F}_p(t)$ and their subrings of $\mathcal{S}$-integral functions. We focus on fields ${\bf K}$ satisfying a local property which we call $q$-boundedness. This can be considered a function field analogue of prior work of the first author which considered algebraic extensions of $\mathbb{Q}$. One simple consequence of our work states that if ${\bf K}$ is a $q$-bounded Galois extension of $\mathbb{F}_p(t)$, then for infinitely many non-constant $u$ the integral closure $\mathcal{O}_{\bf K}$ of $\mathbb{F}_p[u]$ inside ${\bf K}$ is first-order definable in ${\bf K}$. Under the additional assumption that the constant subfield of ${\bf K}$ is infinite, it follows that both $\mathcal{O}_{\bf K}$ and ${\bf K}$ have undecidable first-order theories, and that $\mathbb{F}_p[w]$ is definable in ${\bf K}$ for every non-constant $w$ in ${\bf K}$. Our primary tools are norm equations and the Hasse Norm Principle, in the spirit of Rumely. Our paper has an intersection with a recent arXiv preprint by Mart\'inez-Ranero, Salcedo, and Utreras, although our definability results are more extensive and undecidability results are much stronger.

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arXiv2022

Abelian varieties over finite fields and their groups of rational points

We study the groups of rational points of abelian varieties defined over a finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or, equivalently, whose isogeny classes are determined by squarefree characteristic polynomials. When $\mathrm{End}(A)$ is locally Gorenstein, we show that the group structure of $A(\mathbb{F}_q)$ is determined by $\mathrm{End}(A)$. Moreover, we prove that the same conclusion is attained if $\mathrm{End}(A)$ has local Cohen-Macaulay type at most $ 2$, under the additional assumption that $A$ is ordinary or $q$ is prime. The result in the Gorenstein case is used to characterize squarefree cyclic isogeny classes in terms of conductor ideals. Going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with $N$ rational points in which every abelian group of order $N$ is realized as a group of rational points. Finally, we study when an abelian variety $A$ over $\mathbb{F}_q$ and its dual $A^\vee$ succeed or fail to satisfy several interrelated properties, namely $A\cong A^\vee$, $A(\mathbb{F}_q)\cong A^\vee(\mathbb{F}_q)$, and $\mathrm{End}(A)=\mathrm{End}(A^\vee)$. In the process, we exhibit a sufficient condition for $A\not\cong A^\vee$ involving the local Cohen-Macaulay type of $\mathrm{End}(A)$. In particular, such an abelian variety $A$ is not a Jacobian, or even principally polarizable.

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arXiv2022

Definability and decidability for rings of integers in totally imaginary fields

We show that the ring of integers of $\mathbb{Q}^{\text{tr}}$ is existentially definable in the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$, where $\mathbb{Q}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$ is undecidable and first-order non-definable in $\mathbb{Q}^{\text{tr}}(i)$. More generally, when $L$ is a totally imaginary quadratic extension of a totally real field $K$, we use the unit groups $R^\times$ of orders $R\subseteq \mathcal{O}_L$ to produce existentially definable totally real subsets $X\subseteq \mathcal{O}_L$. Under certain conditions on $K$, including the so-called JR-number of $\mathcal{O}_K$ being the minimal value $\text{JR}(\mathcal{O}_K) = 4$, we deduce the undecidability of $\mathcal{O}_L$. This extends previous work which proved an analogous result in the opposite case $\text{JR}(\mathcal{O}_K) = \infty$. In particular, unlike prior work, we do not require that $L$ contains only finitely many roots of unity.

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arXiv2021

Every finite abelian group is the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$

We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite field $\mathbb{F}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over $\mathbb{F}_q$ when $q$ is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over $\mathbb{F}_2$.

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arXiv2021

Doubly isogenous genus-2 curves with $D_4$-action

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C' are curves over a finite field K, with K-rational base points P and P', and let D and D' be the pullbacks (via the Abel-Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C,P) and (C',P') are *doubly isogenous* if Jac(C) and Jac(C') are isogenous over K and Jac(D) and Jac(D') are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naive heuristics predict, and we provide an explanation for this phenomenon.

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arXiv2020

A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$

For any subset $Z \subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in $L$ such that $C \cap \mathbb{Q}=Z$. Placing a natural topology on the set $\text{Sub}(\overline{\mathbb{Q}})$ of subfields of $\overline{\mathbb{Q}}$, we show that if $Z$ is not thin in $\mathbb{Q}$, then $S_Z$ is meager in $\text{Sub}(\overline{\mathbb{Q}})$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields $L$ have the property that the ring of algebraic integers $\mathcal{O}_L$ is universally definable in $L$. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists$-definable subset of an algebraic extension of $\mathbb Q$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

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arXiv2020

The Structure of the Group of Rational Points of an Abelian Variety over a Finite Field

Let $A$ be a simple abelian variety of dimension $g$ defined over a finite field $\mathbb{F}_q$ with Frobenius endomorphism $\pi$. This paper describes the structure of the group of rational points $A(\mathbb{F}_{q^n})$, for all $n \geq 1$, as a module over the ring $R$ of endomorphisms which are defined over $\mathbb{F}_q$, under certain technical conditions. If $[\mathbb{Q}(\pi) : \mathbb{Q}]=2g$ and $R$ is a Gorenstein ring, then ${A(\mathbb{F}_{q^n}) \cong R/R(\pi^n-1)}$. This includes the case when $A$ is ordinary and has maximal real multiplication. Otherwise, if $Z$ is the center of $R$ and $(\pi^n - 1)Z$ is the product of invertible prime ideals in $Z$, then $A(\mathbb{F}_{q^n})^d \cong R/R(\pi^n - 1)$ where $d = 2g/[\mathbb{Q}(\pi):\mathbb{Q}]$. Finally, we deduce the structure of $A(\overline{\mathbb{F}}_q)$ as a module over $R$ under similar conditions. These results generalize results of Lenstra for elliptic curves.

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arXiv2020

Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of $n$ into odd parts.

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arXiv2019

Undecidability, unit groups, and some totally imaginary infinite extensions of $\mathbb{Q}$

We produce new examples of totally imaginary infinite extensions of $\mathbb{Q}$ which have undecidable first-order theory by generalizing the methods used by Martinez-Ranero, Utreras and Videla for $\mathbb{Q}^{(2)}$. In particular, we use parametrized families of polynomials whose roots are totally real units to apply methods originally developed to prove the undecidability of totally real fields. This proves the undecidability of $\mathbb{Q}^{(d)}_{ab}$ for all $d \geq 2$.

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arXiv2018

Computing the endomorphism ring of an ordinary abelian surface over a finite field

We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal totally real subfield of the endomorphism algebra, generalizing the elliptic curve case.

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