arXiv subjects

Jef Laga

Publications and source records attributed to Jef Laga.

13 recordsLinked to original sources

arXiv2025

Polarizations, torsors and theta groups

Let $\lambda\colon A\rightarrow A^{\vee}$ be a polarization on an abelian variety over a field $k$. If $k$ is not algebraically closed, there might not exist an ample line bundle on $A$ defined over $k$ that represents $\lambda$. To remedy this, Poonen and Stoll have asked the following question: does there exist a line bundle on an $A$-torsor that represents $\lambda$? We give a criterion for the existence of such a torsor and line bundle which only depends on the kernel of $\lambda$. Using this criterion, we show that the answer to the question is yes when the polarization has odd or small even degree. On the other hand, we show that for every $g\geq 7$, there exists a polarized $g$-dimensional abelian variety for which the answer to the question is no.

math.AG↗

arXiv2025

Families of curves in Vinberg representations

Inspired by orbit parametrizations in arithmetic statistics, we explain how to construct families of curves associated to certain nilpotent elements in $\mathbb{Z}/m\mathbb{Z}$-graded Lie algebras, generalizing work of Thorne to the $m\geq 3$ case and the non-simply laced case. We classify such families arising from subregular nilpotents in stable gradings and interpret almost all orbit parametrizations associated with algebraic curves appearing in the literature in this framework. As an extended example, we give a Lie-theoretic proof of the integral orbit parametrization of $5$-Selmer elements of elliptic curves over $\mathbb{Q}$, using a $\mathbb{Z}/5\mathbb{Z}$-grading on a Lie algebra of type $E_8$.

math.NT↗

arXiv2025

Lower bounds on heights of odd degree points of hyperelliptic curves

We develop a reduction theory for the representation of $\mathrm{SL}_n$ on pairs of symmetric $n\times n$ matrices. We apply this theory to the pencils of quadrics arising from divisors on hyperelliptic curves. We use these results to show that, in a density $1$ family, an odd degree point $P$ of degree at most $2g-1$ on the hyperelliptic curve $z^2 = f_0x^{2g+2} + f_1 x^{2g+1} y + \cdots + f_{2g+2}y^{2g+2}$ cannot have small Weil height.

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arXiv2025

Kummers, spinors, and heights

Let $f(x) = x^{2g+1} + c_1 x^{2g} + \dots + c_{2g+1} \in k[x]$ be a polynomial of nonzero discriminant, and let $J$ denote the Jacobian of the odd hyperelliptic curve $C : y^2 = f(x)$. We show that the morphism $J \to \mathbb{P}^{2^g-1}$ associated to the linear system $|2 \Theta|$ may be described explicitly, for any $g \geq 1$, using the theory of pure spinors. We apply this theory to study the heights of rational points in $J(k)$, when $k$ is a number field. As a particular consequence, we show that $100\%$ of monic, degree $2g+1$ polynomials $f(x) \in \mathbb{Z}[x]$ of nonzero discriminant $\Delta(f)$ have the property that, for any non-trivial point $P \in J(\mathbb{Q})$, the canonical height of $P$ satisfies $ \widehat{h}_\Theta(P) \geq \left(\frac{3g-1}{4g(2g+1)} - \epsilon\right) \log | \Delta(f) |$. This is a `density 1' form of the Lang--Silverman conjecture.

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arXiv2024

Vanishing criteria for Ceresa cycles

Let $C$ be a smooth projective curve, and let $J$ be its Jacobian. We prove vanishing criteria for the Ceresa cycle $\kappa(C) \in \mathrm{CH}_1(J)\otimes \mathbb{Q}$ in the Chow group of 1-cycles on $J$. Namely, $(A)$ If $\mathrm{H}_{\mathrm{prim}}^3(J)^{\mathrm{Aut}(C)} = 0$, then $\kappa(C)$ vanishes; $(B)$ If $\mathrm{H}^0(J, \Omega_J^3)^{\mathrm{Aut}(C)} = 0$ and the Hodge conjecture holds, then $\kappa(C)$ vanishes modulo algebraic equivalence. We then study the first interesting case where $(B)$ holds but $(A)$ does not, namely the case of Picard curves $C \colon y^3 = x^4 + ax^2 + bx + c$. Using work of Schoen on the Hodge conjecture, we show that the Ceresa cycle of a Picard curve is torsion in the Griffiths group. Moreover, we determine exactly when it is torsion in the Chow group. As a byproduct, we show that there are infinitely many plane quartic curves over $\mathbb{Q}$ with torsion Ceresa cycle (in fact, there is a one parameter family of such curves). Finally, we determine which automorphism group strata are contained in the vanishing locus of the universal Ceresa cycle over $\mathcal{M}_3$.

math.AG↗

arXiv2024

100% of odd hyperelliptic Jacobians have no rational points of small height

We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of $\mathrm{SO}_{2g+1}$ on self-adjoint $(2g + 1) \times(2g + 1)$-matrices. Using this theory, we show that in a density 1 subset, the Jacobians of these curves have no nontrivial rational points of small height.

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arXiv2024

A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points

Let $\mathcal{F}_g$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over $\mathbb{Q}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_g$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $g\geq 4$, a positive proportion of curves in $\mathcal{F}_g$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $\mathbb{P}^1$.

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arXiv2023

Ceresa cycles of bielliptic Picard curves

We show that the Ceresa cycle $\kappa(C_t)$ of the genus $3$ curve $C_t \colon y^3 = x^4 + 2tx^2 + 1$ is torsion if and only if $Q_t=( \sqrt[3]{t^2 -1},t)$ is a torsion point on the elliptic curve $y^2 = x^3 + 1$. This shows that there are infinitely many smooth plane quartic curves over $\mathbb{C}$ (resp. $\mathbb{Q}$) with torsion (resp. infinite order) Ceresa cycle. Over $\overline{\mathbb{Q}}$, we show that the Beilinson--Bloch height of $\kappa(C_t)$ is proportional to the Neron--Tate height of $Q_t$. Thus, the height of $\kappa(C_t)$ is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls $\kappa(C_t)$ is isomorphic to $\mathfrak{h}^1$ of an appropriate elliptic curve.

math.AG↗

arXiv2023

The geometry and arithmetic of bielliptic Picard curves

We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic multiplication (QM) by the quaternion order of discriminant $6$. This allows us to describe the Galois action on the geometric endomorphism algebra of $P$. As an application, we classify the torsion subgroups of the Mordell-Weil groups $P(\mathbb{Q})$, as both abelian groups and $\text{End}(P)$-modules.

math.AG↗

arXiv2023

Rational torsion points on abelian surfaces with quaternionic multiplication

Let $A$ be an abelian surface over $\mathbb{Q}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur's theorem for elliptic curves, we show that the torsion subgroup of $A(\mathbb{Q})$ is $12$-torsion and has order at most $18$. Under the additional assumption that $A$ is of $\mathrm{GL}_2$-type, we give a complete classification of the possible torsion subgroups of $A(\mathbb{Q})$.

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arXiv2022

Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics

A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne's picture and show that $2$-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are: an analysis of torsors on affine spaces using results of Colliot-Th\'el\`ene and the Grothendieck--Serre conjecture, a study of geometric properties of compactified Jacobians using the Bialynicki-Birula decomposition, and a general construction of integral orbit representatives.

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arXiv2021

Arithmetic statistics of Prym surfaces

We consider a family of abelian surfaces over $\mathbb{Q}$ arising as Prym varieties of double covers of genus-$1$ curves by genus-$3$ curves. These abelian surfaces carry a polarization of type $(1,2)$ and we show that the average size of the Selmer group of this polarization equals $3$. Moreover we show that the average size of the $2$-Selmer group of the abelian surfaces in the same family is bounded above by $5$. This implies an upper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonen and Rains for a family of abelian varieties which are not principally polarized. The proof is a combination of an analysis of the Lie algebra embedding $F_4\subset E_6$, invariant theory, a classical geometric construction due to Pantazis, a study of N\'eron component groups of Prym surfaces and Bhargava's orbit-counting techniques.

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arXiv2020

The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_6$, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.

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