Infinite families of pairs of curves over Q with isomorphic Jacobians
Everett W. Howe
备注：
LaTex, 20 pages. Excluded some degenerate cases from Theorem 2, improved the exposition, simplified some examples, added an application, and included links to Magma code
We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P^1. The first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible Jacobians. The second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple Jacobians. The third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. Examples from these families show that in general it is impossible to tell from the Jacobian of a genus-2 curve over Q whether or not the curve has rational points -- or indeed whether or not it has real points. Our constructions depend on earlier joint work with Franck Leprevost and Bjorn Poonen, and on Peter Bending's explicit description of the curves of genus 2 whose Jacobians have real multiplication by Z[\sqrt{2}].
We construct examples of number fields which are not isomorphic but for which their idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group theory provided by Leonard Scott of a finite group $G$ and subgroups $H_1$ and $H_2$ which are not conjugate in $G$ but for which the $G$-module ${\mathbb Z}[G/H_1]$ is isomorphic to ${\mathbb Z}[G/H_2]$.
We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian varieties are isomorphic to Weil's restriction of quadratic \Q-curves, and present examples.
We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly'', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in R whose minimal polynomials over Q can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugat…
We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t \in K$ and the Galois group of the polynomial $h(x)$ over $K$ is "very big" and $deg(h)$ is an even number >8. (The case of odd $deg(h)>3$ follows easily from previous results of the author.)
We prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian associated with the tropical spectral curve. This result implies that the theta function solution obtained in the authors' previous paper is the complete solution. We also propose a method to solve the initial value problem.
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…
Let A be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over the algebraic closure of k. There exists a canonical extension k' of k, of degree 1 or 2, such that A becomes isomorphic to a Jacobian over k'. The aim of this note is to give a geometric construction of this extension.
In this paper we classify curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12 over an arbitrary field k (of characteristic different from 2 in the D_8 case and from 2 and 3 in the D_{12} case) up to k-isomorphism. As an application of the classification of curves of genus 2 obtained, we get precise arithmetic information on their elliptic quotients and on their jacobians. Over the field k=Q, we show that the elliptic quotients of the curves with automorphisms D_8 and D_{12} are precisely the Q-curves of degrees 2 and 3, respectively, and we determine which curves have jacob…
We compute all the "special" cases of (3,3)-split Jacobians and we parametrize the Igusa-Clebsch invariants of curves of genus two whose Jacobian is (3,3)-isogenous to a product of two elliptic curves from the Hesse pencil.
We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbb A}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$ of the group $\mathrm{Aff}_{2}(K)$ of affine transformations of ${\mathbb A}^{2}$. We then show that the following three statements are equivalent: (i) The Jacobian Conjecture holds in dimension 2; (ii) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are conjugate under $\mathrm{Aut}({\mathbb A}^{2})$; (iii) All …
We study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one-parameter regular local smoothing of a nodal curve $X$, the relative smooth locus of a relative fine compactified Jacobian is isomorphic to the Néron model of the Jacobian of the general fiber, and thus it provides a modular compactification of it. We show that each fine compactified Jacobian of $X$ admits a stratification in terms of certain fine compactified Jacobians of partial normalizations of $X$ and, moreover, that it can be realized as a quotient of the smooth locus of a suitabl…
This paper addresses the following question of Oort: "Are there any postive dimensional locally symmetric subvarieties of the moduli space of abelian varieties that are contained in the jacobian locus and contain the jacobian of at least one smooth curve? Some partial results are given.
Let $K$ be a global field of characteristic different from 2 and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group over $K$ is either the full symmetric group $S_{2g}$ or the alternating group $A_{2g}$. We describe explicitly how to choose (infinitely many) pairs of distinct elements $t_1, t_2$ of $K$ such that the $g$-dimensional jacobian of a hyperelliptic curve $y^2=(x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms over an algebraic closure of $K$ and has big $\ell$-adic monodromy.
Let $K$ be a field of characteristic $p \neq 2$, and let $f(x)$ be a sextic polynomial irreducible over $K$ with no repeated roots, whose Galois group is isomorphic to $\A_5$. If the jacobian $J(C)$ of the hyperelliptic curve $C:y^2=f(x)$ admits real multiplication over the ground field from an order of a real quadratic field $D$, then either its endomorphism algebra is isomorphic to $D$, or $p > 0$ and $J(C)$ is a supersingular abelian variety. The supersingular outcome cannot occur when $p$ splits in $D$.
We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For example, we give examples of surfaces whose Picard numbers jump in rank at all primes of good reduction using Mordell-Weil groups of Jacobians and show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded. The isomorphism result addresses the number fiel…
Given a family of nodal curves, a semistable modification of it is another family made up of curves obtained by inserting chains of rational curves of any given length at certain nodes of certain curves of the original family. We give comparison theorems between torsion-free, rank-1 sheaves in the former family and invertible sheaves in the latter. We apply them to show that there are functorial isomorphisms between the compactifications of relative Jacobians of families of nodal curves constructed through Caporaso's approach and those constructed through Pandharipande's approach.
Let $f:X \to \mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $g\geq 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}\geq 5$ if $k=\mathbb C$ and $g\geq 5$; we also prove that $s_{nc}\geq 4$ if ${\rm char}~k>0$ and the relative Jacobian of $f$ is non-smooth.
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these families admit two fibrations in totally geodesic subvarieties, generalizing a result of Grushevsky and Möller. Countably many of these fibres are Shimura. Thus the Jacobian locus contains infinitely many Shimura subvarieties of positive dimension of any $g \leq 4$.
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 o…
Given a family of intermediate Jacobians (for a polarized variation of Hodge structure of weight -1) on a Zariski-open subset of a complex manifold, we construct an analytic space that naturally extends the family. Its two main properties are: (a) the horizontal and holomorphic sections are precisely the admissible normal functions without singularities; (b) the graph of any admissible normal function has an analytic closure inside our space. As a consequence, we obtain a new proof for the zero locus conjecture of M. Green and P. Griffiths. The construction uses filtered D-modules and M. Saito…
Beauville introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system. In this article, we construct a variant of Beauville's system whose general level set is isomorphic to the complement of the `intersection' of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke.
An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines…
We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a universal sheaf, after an etale base change. Our method consists of considering the functor $F$ of relatively simple, torsion-free, rank 1 sheaves on $X/S$, and showing that certain open subsheaves of $F$ have good properties. Strictly speaking, the functor $F$ is only representable by an algebraic space, but we show that $F$ is representable by a scheme after an etale base change. Finally, we use theta functi…
We present a new technique to study Jacobian variety decompositions using subgroups of the automorphism group of the curve and the corresponding intermediate covers. In particular, this new method allows us to produce many new examples of genera for which there is a curve with completely decomposable Jacobian. These examples greatly extend the list given by Ekedahl and Serre of genera containing such curves, and provide more evidence for a positive answer to two questions they asked. Additionally, we produce new examples of families of curves, all of which have completely decomposable Jacobian…
We introduce two families of ideals, $F$-jumping ideals and $F$-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the $D$-modules $M_α$ that were introduced by Blickle, Mustaţă and Smith. Using strong connections between $F$-jumping ideals and generalized test ideals, we give a characterization of $F$-jumping numbers for hypersurfaces. Furthermore, we give an algorithm that determines whether certain numbers are $F$-jumping numbers. In addition, we use $F$-Jacobian ideals to study intrinsic properties of the singul…
A basic technique for studying a family of Jacobian varieties is to extend the family by adding degenerate fibers. Constructing an extension requires a choice of fibers, and one typically chooses to include either degenerate group varieties or degenerate moduli spaces of sheaves. Here we relate these two different approaches when the base of the family is a regular, 1-dimensional scheme such as a smooth curve. Specifically, we provide sufficient conditions for the line bundle locus in a family of compact moduli spaces of pure sheaves to be isomorphic to the Néron model. The result applies to m…
To every singular reduced projective curve X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting non isomorphic (and even non homeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compa…
Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is …
By tropical Abel-Jacobi theorem, the Jacobian of a tropical curve is isomorphic to the Picard group. A tropical curve in $\mathbb{R}^2$ corresponds to an immersion from a tropical curve to $\mathbb{R}^2$. In this paper, we show that any principal divisor on a tropical curve is the restriction of a principal divisor on the ambient plane $\mathbb{R}^2$.
Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let Λbe its value group. Given a smooth, proper, connected K-curve X and a skeleton Γof the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(Γ) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classic…
We describe the isotypical decomposition of the Jacobian variety JW of the Galois extension W-->T of any fourfold cover of smooth connected irreducible projective complex curves X-->T, in terms of Prym's of intermediate covers. We also compute the degree of the isogenies involved and as a result we obtain new proofs of the bigonal and trigonal constructions. Furthermore, we give examples of families of Jacobians isogenous to a product of Jacobians and of families of Prym varieties isogenous to the product of elliptic curves.
In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the dimension of the locus of hyperelliptic curves of genus $g$ with $p$-rank at most $f$. We also produce families of curves so that the $p$-torsion of the Jacobian of each fibre contains multiple copies of the group scheme $α_p$. The method is to study curves which admit an action by $(\ZZ/2)^n$ so that the quotient is a projective line. As a result, some of these fa…
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.
The compactified Jacobian of any projective curve $X$ is defined as the Simpson moduli space of torsion free rank one degree $d$ sheaves that are semistable with respect to a fixed polarization $H$ on $X$. In this paper we give explicitly the structure of this compactified Simpson Jacobian in the case where $X$ is a generalized tree-like curve, i.e., a projective, reduced and connected curve such that the intersection points of its irreducible components are disconnecting ordinary double points. We prove that it is isomorphic to the product of the compactified Jacobians of a certain degree $d_…
Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of these involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $\mathbb{C}$ …
We construct two families of K3 surfaces over a complex manifold $S$ such that the families are pointwise isomorphic but not locally isomorphic over $S$. This answers a question of Wehler from 1977 in the negative and challenges a more recent result of Meersseman.
We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.
Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture
Université de Strasbourg（斯特拉斯堡大学）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；Saitama University（埼玉大学）；Universität Bielefeld（比勒费尔德大学）
On the Jacobian algebras of Ziegler pairs of plane arrangements
Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学，法国国家科学研究中心，拉雅德实验室）；Simion Stoilow Institute of Mathematics（西蒙·斯托伊洛瓦数学研究所）；University of the National Education Commission Krakow（克拉科夫国民教育委员会大学）
Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $β_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( β_{C'}) = β_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are i…
We construct families of smooth affine surfaces with pairwise non isomorphic A 1-cylinders but whose A 2-cylinders are all isomorphic. These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.
A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian as large as possible, that is, an $\mathbb{F}_p$ vector space of dimension equal to its genus. In this paper we consider the following question: fix a finite field $\mathbb{F}_q$ and a family $\mathscr{F}$ of curves over $\mathbb{F}_q$. Then, what is the probability that a curve in this family is ordinary? We answer this question when $\mathscr{F}$ is either the Artin-Schreier family in any characteristic or a superelliptic family in characteristic 2.
