arXiv subjects
Publications and source records attributed to Reynald Lercier.
We study natural evaluation and interpolation problems for elliptic functions and prove that they allow a recursive treatment using a variant of classical butterflies first introduced by Gauss. We deduce the existence of straight-line programs with complexity scaling with $d\log(d)$ for these problems and present applications to finite field arithmetic, coding theory and cryptography.
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.
We give a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in p-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. These criteria are in the vein of the machinery of "cluster pictures" for hyperelliptic curves. We also construct explicit families of quartic curves that realise all possible stable types, against which we test these criteria. We give numerical examples that illustrate how to use these criteria in practice.
We characterise, in terms of Dixmier-Ohno invariants, the types of singularities that a plane quartic curve can have. We then use these results to obtain new criteria for determining the stable reduction types of non-hyperelliptic curves of genus 3.
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.
We propose an algorithm that calculates isogenies between elliptic curves defined over an extension $K$ of $\mathbb{Q}_2$. It consists in efficiently solving with a logarithmic loss of $2$-adic precision the first order differential equation satisfied by the isogeny. We give some applications, especially computing over finite fields of characteristic 2 isogenies of elliptic curves and irreducible polynomials, both in quasi-linear time in the degree.
We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.
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.
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.
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.
We present an explicit method that, given a generic tuple of Dixmier-Ohno invariants, reconstructs a corresponding plane quartic curve.
In a previous article, we obtained data on the distribution of traces of Frobenius of non-hyperelliptic genus $3$ curves over small finite fields. In the present one, we give a heuristic explanation of these data, by extrapolating from results on the distribution of traces of Frobenius for plane curves whose degree is small with respect to the cardinality of their finite base field. In particular, our methods shed some new light on the asymmetry of the distribution around its mean value, which is related to the Serre obstruction.
We give a minimal system of 476 generators (resp. 510 generators) for the algebra of SL(2,C)-covariant polynomials on binary forms of degree 9 (resp. degree 10). These results were only known as conjectures so far. The computations rely on Gordan's algorithm, and some new improvements.
In this short note, we give an algorithm to get an explicit expression of the Luroth invariant in terms of the Dixmier-Ohno invariants. We also get the explicit factorized expression on the locus of Ciani quartics in terms of the coefficients. Finally, we answer two open questions on sub-loci of singular Luroth quartics.
We show how to speed up the computation of isomorphisms of hyperelliptic curves by using covariants. We also obtain new theoretical and practical results concerning models of these curves over their field of moduli.
This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic model over a degree 2 extension of this field.
We explore parameterizations by radicals of low genera algebraic curves. We prove that for $q$ a prime power that is large enough and prime to $6$, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterized by $3$-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to $2$ modulo $3$. We extend this construction to parameterizations by $\ell$-radicals for small odd integers $\ell$, and make it explicit for $\ell=5$.
We propose a pseudo-primality test using cyclic extensions of $\mathbb Z/n \mathbb Z$. For every positive integer $k \leq \log n$, this test achieves the security of $k$ Miller-Rabin tests at the cost of $k^{1/2+o(1)}$ Miller-Rabin tests.