Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3, and a pair of quaternary quadrics when n = 4. In general, minimal genus one equations of degree n are not unique up to isomorphism. We explain how the number of minimal genus one equations of degree n varies according to the Kodaira symbol of the Jacobian of C. Then we count these equations up to isomorphism over a number field of class number 1.
In this article we give the details of an effective point counting algorithm for genus two curves over finite fields of characteristic three. The algorithm has an application in the context of curve based cryptography. One distinguished property of the algorithm is that its complexity depends quasi-quadratically on the degree of the finite base field. Our algorithm is a modified version of an earlier method that was developed in joint work with Lubicz. We explain how one can alter the original algorithm, on the basis of new theory, such that it can be used to efficiently count points on genus …
We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field (\F_{q}) of large characteristic from (\widetilde{O}(\log^8 q)) to (\widetilde{O}(\log^5 q)). Using our algorithm we compute a 256-bit prime-order Jacobian, suitable for cryptographic applications, and also the order of a 1024-bit Jacobian.
Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real …
We find a closed formula for the number $\operatorname{hyp}(g)$ of hyperelliptic curves of genus $g$ over a finite field $k=\mathbb{F}_q$ of odd characteristic. These numbers $\operatorname{hyp}(g)$ are expressed as a polynomial in $q$ with integer coefficients that depend on the set of divisors of $q-1$ and $q+1$. As a by-product we obtain a closed formula for the number of self-dual curves of genus $g$. A hyperelliptic curve is self-dual if it is $k$-isomorphic to its own hyperelliptic twist.
We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using $p$-adic cohomology.
We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[η]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $\widetilde O((\log q)^6)$ bit-operations, where the constant in the $\widetilde O()$ depends on the ring $\mathbb Z[η]$ and on the degrees of polynomials representing the endomorphism $η$. As a proof-of-concept, we compute the zeta function of a curve defined over a 64-bit prime f…
For any number field, we prove that there exists a stacky curve of genus one half defined over the ring of its integers violating the local-global principle for integral points.
We explain how we computed equations for all genus 4 curves defined of the field with 2 elements, up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.
We prove a genus formula for modular curves of $D$-elliptic sheaves. We use this formula to show that the reductions of modular curves of $D$-elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of $D$ tends to infinity.
We present a table containing the maximal number of rational points on a genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks on Frobenius non-classical quartics over finite fields are given.
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd.
We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any `CM-construction' to produce the required genus-2 curves necessarily takes time exponential in the size of its input. On the other hand, we provide an algorithm for producing a genus-2 curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing…
We show how to calculate the Euler characteristic of a local system associated to an irreducible representation of the symplectic group of genus 3 on the moduli space of curves of genus 3 and the moduli space of principally polarized abelian varieties of dimension 3.
Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})…
We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$g$ hyperelliptic curve defined over $\mathbb F_q$ with explicit real multiplication (RM) by an order $\Z[η]$ in a degree-$g$ totally real number field. It is based on the approaches by Schoof and Pila in a more favorable case where we can split the $\ell$-torsion into $g$ kernels of endomorphisms, as introduced by Gaudry, Kohel, and Smith in genus 2. To deal with these kernels in any genus, we adapt a technique that the author, Gaudry, and Spaenlehauer introduced to model the $\ell$-torsion by stru…
The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.
The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves.
Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension $K$ of the field of $p$-adic numbers $\mathbb{Q}\_p$. The algorithm has a quasi-linear complexity in $\ell$ as well as in the genus of the curves.
Abelian surfaces over finite fields containing no curves of genus $3$ or less
CNRS（法国国家科学研究中心；波尔多大学）；IMB, Université de Bordeaux（阿松森国立大学工程学院）；Facultad de Ingeniería, Universidad Nacional de Asunción（乌得勒支大学数学研究所）；Mathematisch Instituut, Universiteit Utrecht（法属波利尼西亚大学GAATI实验室）；Laboratoire GAATI, Université de la Polynésie française（蔚蓝海岸大学让·亚历山大·迪厄多内实验室）；Laboratoire Jean Alexandre Dieudonnè, Universitè Côte d’Azur
Consider the Jacobian of a hyperelliptic genus two curve defined over a prime field of characteristic p and with complex multiplication. In this paper we show that the p-Sylow subgroup of the Jacobian is either trivial or of order p.
We describe the invariants of plane quartic curves -- nonhyperelliptic genus 3 curves in their canonical model -- as determined by Dixmier and Ohno, with application to the classification of curves with given structure. In particular, we determine modular equations for the strata in the moduli space $\Me_3$ of plane quartics which have at least seven hyperflexes, and obtain an computational characterization of curves in these strata.
We prove that there exists a supersingular nonsingular curve of genus $4$ in arbitrary characteristic $p$. For $p>3$ we shall prove that the desingularization of a certain fiber product over $\mathbb{P}^1$ of two supersingular elliptic curves is supersingular.
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_γ(g)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $γ$. We show that $N_γ(g)=N_γ(3γ)$ for $γ\leq \lfloor g/3\rfloor$ and $N_γ(g)=0$ for $γ> \lfloor 2g/3\rfloor$; thus the question above is true provided that $N_γ(g+1) > N_γ(g)$ for $γ= \lfloor g/3 \rfloor +1, \ldots, \lfloor 2g/3\rfloor$. We also show that $N_γ(3γ)$ coincides with $f_γ$, the number introduced by B…
For affine plane curves defined over the rationals of genus at least two, we show that a decision algorithm for the existence of solutions also yields a search algorithm for all solutions.
We revisit and generalize some geometric techniques behind deterministic primality testing for some integer sequences using curves of genus 1 over finite rings. Subsequently we develop a similar primality test using the Jacobian of a genus 2 curve.
In this paper, we give a universal family of curves of genus 2 whose jacobians have $\sqrt2$ multiplication fixed by the Rosati involution, and several results based on it, including isogenies between jacobians of curves, and jacobians of curves with quaternionic multiplication.
Using class field theory one associates to each curve C over a finite field, and each subgroup G of its divisor class group, unramified abelian covers of C whose genus is determined by the index of G. By listing class groups of curves of small genus one may get examples of curves with many points; we do this for all curves of genus 2 over the fields of cardinality 5,7,9,11,13 and 16, giving new entries for the tables of curves with many points (www.manYPoints.org).
We give a generalization of the method of "Elliptic Curve Chabauty" to higher genus curves and their Jacobians. This method can sometimes be used in conjunction with covering techniques and a modified version of the Mordell-Weil sieve to provide a complete solution to the problem of determining the set of rational points of an algebraic curve $Y$.
Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12).
In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of $L$-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these famili…
A projective, smooth, absolutely irreducible algebraic curve X of genus g defined over a finite field F_q is called optimal if for every other such genus g curve Y over F_q one has $\#Y(F_q)\le \#X(F_q)$. In this paper we show that for $g\le 5$ there is a unique optimal genus g curve over F_2. For g=6 there are precisely two and for g=7 there are at least two.
In this paper we show that the maximum number of rational points possible for a smooth, projective, absolutely irreducible genus 4 curve over a finite field F_7 is 24. It is known that a genus 4 curve over F_7 can have at most 25 points. In this paper we prove that such a curve can have at most 24. On the other hand we provide an explicit example of a genus 4 curve over F_7 having 24 points.
The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields.
The CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining all CM curves of genus $2$ defined over the reflex field. We solve the problem by showing that the list given in Bouyer and Streng [LMS J. Comput. Math., 18(1):507{538, 2015] is complete.
Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We also provide an intrinsic characterization of so-called Legendre elliptic curves.
Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $\text{PGL}_2(\mathbb{F}_q)$ or $\text{PSL}_2(\mathbb{F}_q)$. We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.
In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its jacobian,…
We present a method for computing the Mordell-Weil rank of the jacobian of a curve of genus 2 with multiplication by a square root of 2, based on descent via isogenies of degree 2, and apply it to a family of curves.
In this paper we explicitly compute equations for the twists of all the smooth plane quartic curves defined over a number field k. Since the plane quartic curves are non-hyperelliptic curves of genus 3 we can apply the method developed by the author in a previous article. The starting point is a classification due to Henn of the plane quartic curves with non-trivial automorphism group up to C-isomorphism.
For any number field not containing $\QQ(i),$ we give an explicit construction to prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group is nontrivial.
Based on computational evidence, we formulate a number of conjectures on the distribution of rational points on curves of genus 2 over the rational numbers, in terms of the size of the coefficients of an equation of the form y^2 = f(x) >.
We prove formulae for the countings by orbit of square-tiled surfaces of genus two with one singularity. These formulae were conjectured by Hubert & Lelièvre. We show that these countings admit quasimodular forms as generating functions.
Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus $3$. We find the sharp lower bound for the invariants $φ$, $λ$ and $ε$ for all polarized metrized graphs of genus $3$. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.
In this paper, we consider the hyperelliptic analogue of the Frobenius endomorphism generator and show that it produces sequences with large linear complexity on the Jacobian of genus 2 curves.
