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Christophe Ritzenthaler

Publications and source records attributed to Christophe Ritzenthaler.

At least 19 recordsLinked to original sources
arXiv2026

A complete curve of genus 10 in the moduli space of curves of genus 3 and related questions

We show the existence of a complete trigonal curve of geometric genus 10 in the moduli space $\mathcal{M}_3$ of smooth complex projective curves of genus $3$, and of a complete curve of geometric genus 102 through a general point of $\mathcal{M}_3$. We also show that the moduli space $\mathcal{A}_3$ of principally polarized complex abelian threefolds contains a complete curve whose normalization is a hyperelliptic curve of genus 4.

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arXiv2024

An arithmetic intersection for squares of elliptic curves with complex multiplication

Let $C$ be a genus $2$ curve with Jacobian isomorphic to the square of an elliptic curve with complex multiplication by a maximal order in an imaginary quadratic field of discriminant $-d<0$. We show that if the stable model of $C$ has bad reduction over a prime $p$ then $p \leq d/4$. We give an algorithm to compute the set of such $p$ using the so-called refined Humbert invariant introduced by Kani. Using results from Kudla-Rapoport and the formula of Gross-Keating, we compute for each of these primes $p$ its exponent in the discriminant of the stable model of $C$. We conclude with some explicit computations for $d<100$ and compare our results with an unpublished formula by the third author.

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arXiv2024

Siegel modular forms of degree three and invariants of ternary quartics

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators.

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arXiv2023

Refinements of Katz-Sarnak theory for the number of points on curves over finite fields

This paper goes beyond Katz-Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus $g \geq 3$. The experiments point to a stronger notion of convergence than the one provided by the Katz-Sarnak framework for all curves of genus $\geq 3$. However, for elliptic curves and for hyperelliptic curves of every genus we prove that this stronger convergence cannot occur.

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arXiv2022

Lower bounds on the maximal number of rational points on curves over finite fields

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over ${\mathbb F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result, with a bound of the form $1+q+4 \sqrt{q} -32$ valid for all $g\ge 2$ and for all $q$.

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arXiv2022

Plane quartics over $\mathbb{Q}$ with complex multiplication

We give examples of smooth plane quartics over $\mathbb{Q}$ with complex multiplication over $\overline{\mathbb{Q}}$ by a maximal order with primitive CM type. We describe the required algorithms as we go, these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and reconstruction from these invariants. Finally, we discuss some of the reduction properties of the curves that we obtain.

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arXiv2021

Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus $\leq 4$

Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre's obstruction for principally polarized abelian threefolds isogenous to $E^3$ and of the Igusa modular form in dimension $4$. We illustrate our algorithms with examples of curves with many rational points over finite fields.

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arXiv2020

Reduction type of smooth quartics

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\bar{K}$, in terms of the valuations of certain algebraic invariants of $C$ when the characteristic of the residue field is not $2,\,3,\,5$ or $7$. On the way, we gather several results of general interest on geometric invariant theory over an arbitrary ring $R$ in the spirit of (Seshadri 1977). For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$. We exhibit explicit ones for ternary quartic forms under the action of $\textrm{SL}_{3,R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give simple criteria for the type of reduction.

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arXiv2020

Decomposing Jacobians via Galois covers

Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $ϕ$ with a well-chosen map $Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genus $1$ and $ϕ$ is a degree $3$ map totally ramified over $2$ points.

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arXiv2019

Stable models of plane quartics with hyperelliptic reduction

Let C/K: F = 0 be a smooth plane quartic over a complete discrete valuation field K. In a previous paper the authors togetehr with Q. Liu give various characterizations of the reduction (i.e. non-hyperelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of C: in terms of the existence of a special plane quartic model and in terms of the valuations of the Dixmier-Ohno invariants of C. The last one gives in particular an easy computable criterion for the reduction type. However, it does not produce a stable model, even in the case of good reduction. In this paper we give an algorithm to obtain (an approximation of) the stable model when the reduction of the latter is hyperelliptic and the characteristic of the residue field is not 2. This is based on a new criterion giving the reduction type in terms of the valuations of the theta constants of C. Some examples of the computation of these models are given.

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arXiv2019

Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

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arXiv2018

Principally polarized squares of elliptic curves with field of moduli equal to Q

We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the Generalized Riemann Hypothesis is true, there are no further examples of such curves. More generally, we prove under the Generalized Riemann Hypothesis that there exist exactly 46 genus-2 curves over $\overline{\mathbb{Q}}$ with field of moduli $\mathbb{Q}$ whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order.

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arXiv2018

On the Prym variety of genus 3 covers of genus 1 curves

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate case of a result by Nils Bruin.

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