arXiv subjects

Valentijn Karemaker

Publications and source records attributed to Valentijn Karemaker.

At least 19 recordsLinked to original sources

arXiv2026

Oort's conjecture on supersingular abelian varieties in odd characteristic

Oort's conjecture asserts that for any $g \geq 2$ and any prime $p$, every geometric generic member in the supersingular locus $\mathcal{S}_g$ has automorphism group $\{ \pm 1 \}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construct, for any $g \geq 3$, a closed subvariety of dimension $g-1$ which contains an open dense subset $\mathcal{U}$ of $a$-invariant $g-2$ such that $\mathcal{U}$ meets every irreducible component of $\mathcal{S}_g$ and every geometric point in $\mathcal{U}$ has automorphism group $\{ \pm 1 \}$ if $p >2$. This gives an independent proof of Oort's conjecture for $p>2$. When $g>3$, so $a = g-2 > 1$, this provides complementary information to the $a=1$ locus investigated in Viehmann's work on how the automorphism groups interact with the geometry.

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arXiv2026

Corrigendum to "Isomorphism classes of Drinfeld modules over finite fields"

In this note we provide corrections to Theorem 5.4 of the paper ``Isomorphism classes of Drinfeld modules over finite fields'', arXiv:2209.15033. The main theorems of this paper, Theorem A and B in its introduction, are valid as stated; in the proof of Theorem B the argument needs to be modified by replacing the erroneous Theorem 5.4 by the theorem of this note.

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arXiv2024

Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a fractional $W\otimes_{\mathbb Z_p} \mathbb Z_p[\pi,q/\pi]$-ideal, with $\pi$ the Frobenius endomorphism and $W$ the ring of integers in an unramified extension of $\mathbb Q_p$ of degree $a$. The latter ideal should be compatible at $p$ with the former and stable under the action of a semilinear Frobenius (and Verschiebung) operator; it will be the Dieudonn\'e module of the corresponding abelian variety. Using this categorical description we create effective algorithms to compute isomorphism classes of these objects and we produce many new examples exhibiting exotic patterns.

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arXiv2024

Supersingular Ekedahl-Oort strata and Oort's conjecture

Let $\mathcal{A}_g$ be the moduli space over $\overline{\mathbb{F}}_p$ of $g$-dimensional principally polarised abelian varieties, where $p$ is a prime. We show that if $g$ is even and $p\geq 5$, then every geometric generic member in the maximal supersingular Ekedahl-Oort stratum in $\mathcal{A}_g$ has automorphism group $\{ \pm 1\}$. This confirms Oort's conjecture in the case of $p\geq 5$ and even $g$. We also separately prove Oort's conjecture for $g=4$ and any prime $p$.

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arXiv2024

Uniqueness of indecomposable idempotents in algebras with involution

We prove uniqueness of a decomposition of $1$ into indecomposable Hermitian idempotents in an order of a finite-dimensional $\mathbb{Q}$-algebra with positive involution, by generalising a result of Eichler on unique decomposition of lattices. We use this result to prove that polarised abelian varieties over any field admit a unique decomposition into indecomposable polarised abelian subvarieties, a result previously shown by Debarre and Serre with different methods and over algebraically closed fields. We prove that an analogous uniqueness result holds true for arbitrary polarised integral Hodge structures, and derive a consequence for their automorphism groups.

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arXiv2023

A survey of local-global methods for Hilbert's Tenth Problem

Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each $f\in\mathbb{Z}[X_{1},\dots,X_{n}]$, whether the diophantine equation $f(X_{1},...,X_{n})=0$ has a solution in R. The celebrated `Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for $\mathbb{Z}$ is unsolvable, i.e.~there is no such algorithm. Since then, Hilbert's Tenth Problem has been studied in a wide range of rings and fields. Most importantly, for {number fields and in particular for $\mathbb{Q}$}, H10 is still an unsolved problem. Recent work of Eisentr\"ager, Poonen, Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward what is known in this area, and has made essential use of local-global principles for quadratic forms, and for central simple algebras. We give a concise survey and introduction to this particular rich area of interaction between logic and number theory, without assuming a detailed background of either subject. We also sketch two further directions of future research, one inspired by model theory and one by arithmetic geometry.

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arXiv2022

Isomorphism classes of Drinfeld modules over finite fields

We study isogeny classes of Drinfeld $A$-modules over finite fields $k$ with commutative endomorphism algebra $D$, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order $A[\pi]$ of $D$ occurs as an endomorphism ring by proving when it is locally maximal at $\pi$, and show that this happens if and only if the isogeny class is ordinary or $k$ is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring $\mathcal{E}$ of a Drinfeld module $\phi$ up to $D$-linear equivalence acts on the isomorphism classes in the isogeny class of $\phi$, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases.

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arXiv2022

When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group?

We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$.

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arXiv2021

Polarizations of abelian varieties over finite fields via canonical liftings

We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical liftings to characteristic zero, i.e., a lifting for which the reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between abelian varieties over finite fields and fractional ideals in \'etale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations.

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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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arXiv2020

A comparison between obstructions to local-global principles over semiglobal fields

We consider local-global principles for rational points on varieties, in particular torsors, over one-variable function fields over complete discretely valued fields. There are several notions of such principles, arising either from the valuation theory of the function field, or from the geometry of a regular model of the function field. Our results compare the corresponding obstructions, proving in particular that a local-global principle with respect to valuations implies a local-global principle with respect to a sufficiently fine regular model.

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arXiv2020

Dynamical Belyi maps and arboreal Galois groups

We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of their Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. This allows us to study the behavior of the monodromy groups under specialization of the maps, and to derive applications to dynamical sequences.

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arXiv2020

Cubic function fields with prescribed ramification

This article describes cubic function fields $L/K$ with prescribed ramification, where $K$ is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure $K'/K$ of $L/K$ is of genus zero, and a description of the twists of $L/K$ up to isomorphism over $K$. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows M\"obius transformations on $K$. The article concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

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arXiv2020

Mass formula and Oort's conjecture for supersingular abelian threefolds

Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field $k$ of characteristic $p$. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic three-dimensional principally polarised supersingular abelian variety over $k$ of characteristic $\neq 2$ has automorphism group $\{ \pm 1 \}$.

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arXiv2017

Fully maximal and fully minimal abelian varieties

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in characteristic $2$.

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arXiv2017

Dynamical Belyi maps

We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on the rational dynamics.

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arXiv2016

Large Galois images for Jacobian varieties of genus 3 curves

Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation ρ_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l).

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arXiv2015

Hecke algebras for $\mathrm{GL}_n$ over local fields

We study the local Hecke algebra $\mathcal{H}_{G}(K)$ for $G = \mathrm{GL}_n$ and $K$ a non-archimedean local field of characteristic zero. We show that for $G = \mathrm{GL}_2$ and any two such fields $K$ and $L$, there is a Morita equivalence $\mathcal{H}_{G}(K) \sim_M \mathcal{H}_{G}(L)$, by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for $G = \mathrm{GL}_n$, there is an algebra isomorphism $\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)$ which is an isometry for the induced $L^1$-norm if and only if there is a field isomorphism $K \cong L$.

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