Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non hyperelliptic locus M_3^{nh} of the moduli space M_3 of curves of genus 3. By adding to these computations the knowed results on the hyperelliptic locus we obtain a complete picture of these strata for M_3.
In this paper we generalize the formula of Frobenius-Stickelberger and the formula of Kiepert to genus-three case. The latter is well-known determinant expression for any division polynomial of any elliptic curve.
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.
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 describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in $C(\mathbb{Q})$.
We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of $p$-adic integrals, which is known to be finite and include the set of rational points $C(\mathbb{Q})$. We implemented an algorithm in Sage to carry out the Chabauty-Coleman method on a database of $5870$ curves.
In arithmetic and algebraic geometry, superspecial (s.sp.\ for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If $g \geq 4$, it is not even known whether there exists such a curve of genus $g$ in general characteristic $p > 0$, and in the case of $g=4$, several computational approaches to search for those curves have been proposed. In the genus-$4$ hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp.\ curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case w…
Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng, we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus 3 hyperelliptic curves over a finite field $\mathbb{F}_p$ with a …
We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points ov…
In $2000$, Colliot-Thélène and Poonen showed how to construct algebraic families of genus one curves violating the Hasse principle. Poonen explicitly constructed an algebraic family of genus one cubic curves violating the Hasse principle using the general method developed by Colliot-Thélène and himself. The main result in this paper generalizes the result of Colliot-Thélène and Poonen to arbitrarily high genus hyperelliptic curves. More precisely, for $n > 5$ and $n \not\equiv 0 \pmod{4}$, we show that there is an algebraic family of hyperelliptic curves of genus $n$ that is counterexamples to…
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered.
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not su…
We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also show that, if we want such a curve to be just geometrically hyperelliptic, so having a degree two map to a conic, then there are some with genus 3.
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and…
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 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 describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to $(\ZZ/2\ZZ)^3$ (over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3. These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of character…
We develop an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field $k$ of characteristic $\neq 2$. In particular, we provide explicit equations defining the Kummer variety $\mathcal K$ as a subvariety of $\mathbb P^7$, together with explicit polynomials giving the duplication map on $\mathcal K$. A careful study of the degenerations of this map then forms the basis for the development of an explicit theory of heights on such Jacobians when $k$ is a number field. We use this input to obtain a good bound on the difference between naive an…
Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton d…
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 t…
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.
A recursion relation of hyperelliptic psi functions of genus two, which was derived by D.G. Cantor (J. reine angew. Math. 447 (1994) 91-145), is studied. As Cantor's approach is algebraic, another derivation is presented as a natural extension of the analytic derivation of the recursion relation of the elliptic psi function.
We consider the problem of efficient computation in the Jacobian of a hyperelliptic curve of genus 3 defined over a field whose characteristic is not 2. For curves with a rational Weierstrass point, fast explicit formulas are well known and widely available. Here we address the general case, in which we do not assume the existence of a rational Weierstrass point, using a balanced divisor approach.
In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of these curves, when ordered by height, is 3. In this paper, we consider the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational non-Weierstrass point. We show that when these curves are ordered by height, the average size of the 2-Selmer group of their Jacobians is 6. This yields an upper bound of 5/2 on the average rank of the Mo…
We establish a connection between torsion packets on curves of genus $2$ and pairs of elliptic curves realized as double covers of the projective line $\mathbb{P}_{x}^{1}$ that have many common torsion $x$-coordinates. This can be used to show that the set of common torsion $x$-coordinates has size at least $22$ infinitely often and has $34$ elements in some cases. We also explain how we obtained the current record example of a hyperelliptic torsion packet on a genus $2$ curve.
It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford's classic "Lectures on Theta", where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, …
In this paper we prove that there are no hyperelliptic supersingular curves over F_2bar of genus 2^n-1 for any integer n>1. Let g be a natural number, and h=floor(log_2(g+1)+1). Let X be a hyperelliptic curve over F_2bar of genus g>2 and 2-rank zero, given by an affine equation y^2-y=c_{2g+1} x^{2g+1} +...+ c_1 x. We prove that the first slope of the Newton polygon of X is bigger than or equal to 1/h. We also prove that the equality holds if (I) g<2^h-2, c_{2^h-1} is nonzero; or (II) g=2^h-2, c_{2^h-1} or c_{3(2^{h-1})-1} is nonzero. We prove that genus-4 hyperelliptic curve over F_2bar are pr…
In this paper we prove new explicit formulas for Faltings' $δ$-invariant of an arbitrary hyperelliptic Riemann surface. This has several applications: For example we obtain an explicit lower bound for $δ$ depending only on the genus, and we deduce new explicit bounds for the Arakelov self-intersection number $ω^2$ associated to hyperelliptic curves over number fields. Furthermore, we obtain an improved version of Szpiro's small points conjecture for hyperelliptic curves of genus at least $3$. Our method allows us in addition to establish a generalization of Rosenhain's formula on $θ$-derivativ…
The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Leprévost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with non-reduced components in the special fibre of their Néron model. In this present paper we explain how the real period can be computed, and how the verification has been extended to many more hyperelliptic curves, some of genus 3, 4 and 5, without using modular methods.
In this paper we consider the question of whether there exists a hyperelliptic curve of genus $g$ which is defined over $\FF_q$ but has no rational points over $\FF_q$ for various pairs $(g,q)$.
Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.
We prove the non-existence of Shimura subvarieties of positive dimension contained generically in the hyperelliptic Torelli locus for curves of genus at least 8, which is an analogue of Oort's conjecture in the hyperelliptic case.
Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N. The algorithm relies on an adaptation of the "accumulating remainder tree" to matrices with entries in a quadratic field. We report on an implementation, and compare its performance to previous algorithms for the ordinary hyperelliptic case.
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.
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})…
For a given hyperelliptic curve $C$ over a finite field with Jacobian $J_C$, we consider the hyperelliptic analogue of the congruential generator defined by $W_n=W_{n-1}+D$ for $n\geq 1$ and $D,W_0\in J_C$. We show that curves of genus 2 produce sequences with large linear complexity.
We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold fo…
This paper determines the normal forms of hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 0 < g< 9. We also show that every hyperelliptic supersingular curve of genus 9 over F has an equation y^2-y=x^19+c^8x^9+c^3x for some c in F_2bar. The paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus g < 10 over F_2bar.
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their…
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…
This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$, and an algorithm to compute the automorphism group of a (not necessarily superspecial) hyperelliptic curve over finite fields. The first algorithm works for any $(g,q)$ such that $q$ and $2g+2$ are coprime and $q>2g+1$. As an application, we enumerate superspecial hyperelliptic curves of genus $g=4$ over $\mathbb{F}_{p}$ for $11 \leq p \leq 23$ and over $\math…
We construct three-dimensional families of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, 7, 10, 20, and 30, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian. We show that the Jacobians are generically absolutely simple, and describe the kernels of the isogenies. The families are derived from Cassou--Noguès and Couveignes' explicit classification of pairs $(f,g)$ of polynomials such that $f(x_1) - g(…
We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$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…
By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted projective space P(1,1,g+1) via an equation of the form C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the coefficients f_i lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all hyperelliptic curves over Q of genus g by height. Then we prove that, as g tends t…
Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve $\mathcal{C}$ of genus $g\geq 2$ has level $2$. We provide a good number of examples and raise a conjecture.
The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\sqrt{q}$ is distributed as the trace of a random $2g\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\pm 1$ with p…
The Zeta function of a curve $C$ over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix $Θ_C$. Following the work of Rudnick, we compute the expected value of $\mbox{tr}(Θ_C^n)$ over the moduli space of hyperelliptic curves of genus $g$, over a fixed finite field $\mathbb{F}_q$, in the limit of large genus. As an application, we compute the expected value of the number of points on $C$ in $\mathbb{F}_{q^n}$ as the genus tends to infinity. We also look at biases in both expected values for small values of $n$.
We study the averages of products of traces of high powers of the Frobenius class of hyperelliptic curves of genus g over a fixed finite field. We show that for increasing genus g, the limiting expectation of these products equals to the expectation when the curve varies over the unitary symplectic group USp(2g). We also consider the scaling limit of linear statistics for eigenphases of the Frobenius class of hyperelliptic curves, and show that their first few moments are Gaussian.
We introduce the notion of isolated genus two curves. As there is no known efficient algorithm to explicitly construct isogenies between two genus two curves with large conductor gap, the discrete log problem (DLP) cannot be efficiently carried over from an isolated curve to a large set of isogenous curves. Thus isolated genus two curves might be more secure for DLP based hyperelliptic curve cryptography. We establish results on explicit expressions for the index of an endomorphism ring in the maximal CM order, and give conditions under which the index is a prime number or an almost prime numb…
It is known that for a smooth hyperelliptic curve to have a large $a$-number, the genus must be small relative to the characteristic of the field, $p>0$, over which the curve is defined. It was proven by Elkin that for a genus $g$ hyperelliptic curve $C$ to have $a_C=g-1$, the genus is bounded by $g<\frac{3p}{2}$. In this paper, we show that this bound can be lowered to $g <p$. The method of proof is to force the Cartier-Manin matrix to have rank one and examine what restrictions that places on the affine equation defining the hyperelliptic curve. We then use this bound to summarize what is kn…
