Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weil- and the Tate-pairing are non-degenerate over the same field extension of the ground field.
From this generalization we get a complete description of the l-torsion subgroups of Jacobians of supersingular genus two curves. In particular, we show that for l>3, the l-torsion points are rational over a field extension of degree at most 24.
We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety.
We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Vélu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.
We prove that the geometric genus p of a curve in a very generic Jacobian of dimension g>3 satisfies either p=g or p>2g-3. This gives a positive answer to a conjecture of Naranjo and Pirola. For low values of g the second inequality can be further improved to p>2g-2.
The aim of this note is to describe the restriction map from the moduli space of stable rank 2 bundle with small $c_2$ on a jacobian $X$ of dimension 2, to the moduli space of stable rank 2 bundles on the corresponding genus 2 curve $C$ embedded in $X$.
We prove that in most cases relevant to cryptography, the Frobenius endomorphism on the Jacobian of a genus two curve is represented by a diagonal matrix with respect to an appropriate basis of the subgroup of l-torsion points. From this fact we get an explicit description of the Weil-pairing on the subgroup of l-torsion points. Finally, the explicit description of the Weil-pairing provides us with an efficient, probabilistic algorithm to find generators of the subgroup of l-torsion points on the Jacobian of a genus two curve.
Given a canonical genus three curve $X=\{F=0\}$, we construct, emulating Mumford discussion for hyperelliptic curves, a set of equations for an affine open subset of the jacobian $JX$. We give explicit algorithms describing the law group in $JX$. Finally we introduce a related construction by means of an imbedding of the open set previously described in a Grassmanian variety.
For a nonsingular projective curve $C$ of genus 3 defined over an algebraically closed field of characteristic $p > 2$, we give a necessary and sufficient condition that the Jacobian variety $J(C)$ has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.
L. Moret-Bailly constructed families $\mathfrak{C}\rightarrow \mathbb{P}^1$ of genus 2 curves with supersingular jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family.
We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained withi…
A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimension…
In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.
Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves with complex multiplication. In particular, we show that the Weil- and the Tate-pairing on such a Jacobian are non-degenerate over the same field extension of the ground field.
We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational point of order 48, parametrized by a rank-2 elliptic curve over Q, and we exhibit a single genus-2 curve over Q whose Jacobian has a rational point of order 70, the largest order known. We also give new examples of genus-2 Jacobians with rational points of order 27, 28, and 39. Most of our examples are produced by `gluing' two elliptic curves together along t…
For each universal genus-$g$ polarization $μ$ of degree $d$, we construct a universal tropical Jacobian $J_{μ,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{μ,g}^{trop}$. In particular, we prove that the natural compactification of $J_{μ,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.
In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac} (X))\otimes Q$ contains the totally real cubic number field $Q(ζ_7 +\barζ_7 )$. We construct explicit three-dimensional families whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when $X$ is hyperelliptic was studied in a previous work by Hoffman and Wang and some nonhyperelliptic curves were constructed in a previous paper by Hoffman, Z. Liang. Sakai and Wang.
In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac}(X)) \otimes Q$ contains the totally real cubic number field $Q(ζ_ 7 + \overlineζ_7)$. We construct explicit two-dimensional families defined over $Q(s; t)$ whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied by the authors Hoffman and Wang in a previous work. We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.
We give a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix, and derive some identities between Jacobian Nullwerte in dimension three.
We study the singular locus on the algebraic surface $§_n$ of genus 2 curves with a $(n, n)$-split Jacobian. Such surface was computed by Shaska in \cite{deg3} for $n=3$, and Shaska at al. in \cite{deg5} for $n=5$. We show that the singular locus for $n=2$ is exactly th locus of the curves of automorphism group $D_4$ or $D_6$. For $n=3$ we use a birational parametrization of the surface $§_3$ discovered in \cite{deg3} to show that the singular locus is a 0-dimensional subvariety consisting exactly of three genus 2 curves (up to isomorphism) which have automorphism group $D_4$ or $D_6$. We furt…
We give an algorithm to compute $(\ell,\ell,\ell)$-isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve.
We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves $E_1$ and $E_2$ is a 2-dimensional subvariety of the hyperelliptic moduli $\mathcal H_3$. We determine this subvariety explicitly. For any given moduli point $\mathfrak p \in \mathcal H_3$ we determine explicitly if the corresponding genus 3 curve $\mathcal X$ belongs or not to such family. When it does, we can determine elliptic subcovers $E$, $E_1$, and $E_…
In this paper we establish that the singularities of the universal compactified Jacobian are canonical if the genus is at least four. As a corollary we determine the Kodaira dimension and the Iitaka fibration of the universal compactified Jacobian for every degree and genus. We also determine the birational automorphism group for every degree if the genus is at least twelve. This extends work of G. Farkas and A. Verra, as well as that of G. Bini, C. Fontanari and the third author.
In this paper we study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure (IVHS) is of rank 1. We show that if the genus g is greater or equal to 8 or g=6,7 and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete a result of Naranjo and Pirola. We show in fact that if the genus g is greater or equal to 6, the hyperelliptic locus is the only 2g-1-dimensional sub-locus Y of the moduli space of curves of genus g, such that for the general element [C] in Y, its Ja…
Let $C$ be a curve of genus 2 and $ψ_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $ϕ_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering. We determine all possible ramifications for $ϕ_1$. If $ψ_1:C \lar E_1$ is maximal then there exists a maximal map $ψ_2:C\lar E_2$, of degree $n$, to some elliptic curve $E_2$ such that there is an isogeny of degree $n^2$ from the Jacobian $J_C$ to $E_1 \times E_2$. We say that $J_C$ is $(n,n)$-decomposable. If the degree $n$ is odd the pair $(ψ_2, E_2)$ is canonic…
Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.
In this paper we construct infinitely many Shimura curves contained in the locus of Jacobians of genus four curves. All Jacobians in these families are ${\mathbb Z}/3$ covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura-Teichmuller curve that parameterizes Jacobians that are suitable ${\mathbb Z}/6$ covers of ${\mathbb P}^1$. We compute explicitly the period matrices of the Shimura curves we construct using the original construction of Shimura for moduli spaces of abelian varieties with automorphisms.
Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryptography, etc. Choosing genus 2 curves suitable for such applications is an important step of such algorithms. In existing algorithms often such curves are chosen using equations of moduli spaces of curves with decomposable Jacobians or Humbert surfaces. In these lectures we will cover basic properties of genus 2 curves, moduli spaces…
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.
We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus 1 and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus 1 can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of ge…
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 genu…
We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1, 3)-polarised abelian surface. We investigate these curves and show that their Jacobians contain a surface and its dual as complementary abelian subvarieties.
This paper is devoted to constructing an explicit efficient representation for the Jacobian variety of a nonsingular curve of genus greater than 1, and its group law. We describe an algorithm for executing the group law on the Jacobian elements, consisting of several steps. Finally we exhibit two examples. The first example exhibits an explicit formulation for the Jacobian of a well known non singular elliptic curve of genus 1, that is, the Weierstrass curve. The second example introduces an explicit computation of the group law for a non singular curve of genus 3, $C : ω_2^4 = ω_1 (ω_1-ω_0)(ω…
In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the formula given in [5] works only under certain conditions.
In this paper we obtain conditions on the divisors of the group order of the Jacobian of a hyperelliptic genus 2 curve, generated by the complex multiplication method described by Weng (2003) and Gaudry (2005). Examples, where these conditions imply that the Jacobian has a large cyclic subgroup, are given.
We show that the moduli spaces of non-triple cyclic coverings of genus 2 curves and that of even spin curves of genus 2 are birationally isomorphic via the Prym map. We describe the log canonical models of the moduli space of even spin curves of genus 2 and use this to extend the above birational map.
We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressi…
We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs $(m,n)$ the locus of polarised abelian surfaces of type $(1,d)$ that contain two complementary elliptic curve of exponents $m,n$, denoted $\mathcal{E}_d(m,n)$ is non-empty. We show that if $d$ is square-free, the locus $\mathcal{E}_d(m,n)$ is an irreducible surface (if non-empty). We also show that the loci $\mathcal{E}_d(d,d)$ can have many components if $d$ is an odd square. As an application, we show that for a genus $3$ curve with a completely decomposable Jacobian (i.e. isogenous to a pr…
In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.
We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor.
We give a criterion to distinguish between a genus three Jacobian and its [-1] twist in terms of the product of the 36 even theta nulls. We also express the product of the 36 theta nulls in terms of the discriminant of a genus three curve. The results are arithmetic in nature and thus add to previous work over C on the product of the even theta nulls. They generalize previous work of Ritzenthaler and Lachaud for Abelian threefolds which are (2,2,2) isogenous to a product of elliptic curves. This answers a 2003 of letter of Jean-Pierre Serre to Jaap Top.
Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then…
We present an efficient endomorphism for the Jacobian of a curve $C$ of genus 2 (hyperelliptic) for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulae for divisor doubling and addition of divisors with disjoint support in $\mathbb{J}(C)$ using only base field operations. Explicit formulae is presented for this third case and a slightly different approach for divisor doubling.
In this paper we study genus 2 curves whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies …
Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get …
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 …
In recent years there has been an interest in constructing examples of closed Riemann surfaces whose jacobian varieties are isogenous to a product of many elliptic factors and some other jacobian varieties. The first ones, provided by Ekedahl and Serre, are examples for which the isogenous decomposition has all factors being elliptic curves. It is well known that given two elliptic curves $E_{1}$ and $E_{2}$, there is a closed Riemann surface $X$ of genus two, with equations in terms of the elliptic curves, and whose jacobian variety $JX$ is isogenous to $E_{1} \times E_{2}$. In this paper, gi…
We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.
We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include an example of $A_f$ with quaternionic multiplication for which we find numerically a curve $C$ whose Jacobian is $A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $A_f$.
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 this paper we study the irreducible components of the compactified Jacobian of a ribbon $X$ of arithmetic genus $g$ over a smooth curve $X_{\mathrm{red}}$ of genus $\bar{g}$. We prove that when $g\geq 4\bar{g}-2$ the moduli space of rank $2$ semistable vector bundles over $X_{\mathrm{red}}$ is not an irreducible component and we determine the irreducible components in which it is contained. This answers a question of D. Chen and J.L. Kass in [CK] and completes their results.
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 consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fi…
We discuss the theory of generalized Weierstrass $σ$ and $\wp$ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the "purely trigonal" (or "cyclic trigonal") curve $y^3=x^5+λ_4 x^4 +λ_3 x^3+λ_2 x^2 +λ_1 x+λ_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the $\wp$ functions, and the derivation of an addition formulae.
Let $\C$ be a genus 2 curve defined over $k$, $char (k) =0$. If $\C$ has a $(3,3)$-split Jacobian then we show that the automorphism group $Aut(\C)$ is isomorphic to one of the following: $\bZ_2, V_4, D_8$, or $D_{12}$. There are exactly six $\bC$-isomorphism classes of genus two curves $\C$ with $Aut(\C)$ isomorphic to $D_8$ (resp., $D_{12}$). %We compute their absolute invariants $i_1, i_2, i_3$. We show that exactly four (resp., three) of these classes with group $D_8$ (resp., $D_{12}$) have representatives defined over $\bQ$. We discuss some of these curves in detail and find their rationa…
