We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3^n) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of the criterion are provided; the first relies on a theorem of Shimura, while the second uses the moduli interpretation.
We give a structure theorem for the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F(x)$. We study existence of torsion on curves of the form $y^q=x^p-x+a$ over finite fields of characteristic $p$. We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over $\mathbb Q$.
We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.
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…
Let $ρ$ be a mod $\ell$ Galois representation. We show how to compute $ρ$, given the characteristic polynomial of the image of the Frobenius at one prime $p$ and a curve $C$ whose Jacobian contains $ρ$ in its $\ell$-torsion. The main ingredient is a method to $p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.
Suppose $X$ is a smooth, proper, geometrically connected curve over $\mathbb F_q$ with an $\mathbb F_q$-rational point $x_0$. For any $\mathbb F_q^{\times}$-character $σ$ of $π_1(X)$ trivial on $x_0$, we construct a functor $\mathbb L_n^σ$ from the derived category of coherent sheaves on the moduli space of deformations of $σ$ over the Witt ring $W_n(\mathbb F_q)$ to the derived category of constructible $W_n(\mathbb F_q)$-sheaves on the Jacobian of $X$. The functors $\mathbb L_n^σ$ categorify the Artin reciprocity map for geometric class field theory with $p$-torsion coefficients. We then giv…
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 prove that Néron models of jacobians of generically-smooth nodal curves over bases of arbitrary dimension are quasi-compact (hence of finite type) whenever they exist. We give a simple application to the orders of torsion subgroups of jacobians over number fields.
We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P^1 of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X^4 + Y^4 + Z^4) - 96914(X^2 Y^2 + X^2 Z^2 + Y^2 Z^2) = 0, whose Jacobian has 864 rational torsion points. This paper has appeared in Forum Math. 12 (2000) 315-364.
We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation $y^2=x^{2g+1}+x$ with $g \geq 2$. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.
In this paper, we describe the construction of superelliptic curves with a rational point of prescribed order on their jacobians. The construction is based on Hensel's Lemma and produces for a given integer $N$ a superelliptic curve of genus linear in $N$ with a rational $N$-division point on the jacobian. The method is illustrated with multiple examples.
Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the $p$-adic loop group and the formal group.
We study the rational torsion subgroup of the modular Jacobian $J_0(N)$ for $N$ a square-free integer. We give a new proof of a result of Ohta on a generalization of Ogg's conjecture: for a prime number $p \nmid 6N$, the $p$-primary part of the rational torsion subgroup equals that of the cuspidal subgroup. Whereas previous proofs of this result used explicit computations of the cardinalities of these groups, we instead use their structure as modules for the Hecke algebra.
Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points on $J_0(3p)$ and the cuspidal group of $J_0(3p)$, respectively. We prove that the $3$-primary subgroups of $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ coincide unless $p\equiv 1 \pmod 9$ and $3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}$.
For any positive integer $N$, we prove that the rational torsion subgroup of $J_0(N)$ agrees with its rational cuspidal subgroups up to a factor of $6N\prod_{p\mid N}(p^2-1)$. Moreover, for modular Jacobians of the form $J_0(DC)$ with $D$ a positive square-free integer and $C$ any positive divisor of $D$, we prove that the $ψ$-part of the torsion subgroup of $J_0(DC)$ agrees with the $ψ$-part of its cuspidal subgroup up to a factor of $6D\prod_{p\mid D}(p^2-1)$, where $ψ$ is any quadratic character of conductor dividing $C$.
Let $D$ be an odd square-free positive integer and $C$ a divisor of $D$. For any quadratic character $χ$ modulo $C$, we prove that the $χ$-part of the group $J_0(DC)_\text{tor}$ of torsion points of $J_0(DC)$ coincides with the $χ$-part of its cuspidal subgroup, away from those primes of bad reduction or where possible congruences between oldforms and newforms occur.
We consider the generalised Jacobian $J_{0}(N)_{\mathbf{m}}$ of the modular curve $X_{0}(N)$ of level $N$, with respect to the modulus $\mathbf{m}$ consisting of all cusps on the modular curve. When $N$ is odd, we determine the group structure of the rational torsion $J_{0}(N)_{\mathbf{m}}(\mathbb{Q})_{tor}$ up to $2$-primary and $l$-primary parts for any prime $l$ dividing $N$. Our results extend those of Wei--Yamazaki for squarefree levels and Yamazaki--Yang for prime-power levels.
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.
Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the $\ell$-primary part of the cuspidal divisor class group for any prime $\ell$ not dividing $q-1$.
Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian variety over $\mathbb{Q}$. For an odd prime $\ell$, we provide sufficient conditions for the non-existence of rational points of order $\ell$ on $J$. As an application, we find some non-trivial subgroups of the kernel of an isogeny from the new quotient of $J_0(pq)$ to $J$.
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 $\mathbb{C}$ …
For $m \in {\mathbb N}$, let $S_m$ be the Suzuki curve defined over ${\mathbb F}_{2^{2m+1}}$. It is well-known that $S_m$ is supersingular, but the p-torsion group scheme of its Jacobian is not known. The a-number is an invariant of the isomorphism class of the p-torsion group scheme. In this paper, we compute a closed formula for the a-number of $S_m$ using the action of the Cartier operator on $H^0(S_m,Ω^1)$.
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…
Given a principally polarized abelian variety $A$ of dimension $g$ over an algebraically closed field $k$ of characteristic $p$, the $p$ torsion $A[p]$ is a finite flat $p$-torsion group scheme of rank $p^{2g}$. There are exactly $2^g$ possible group schemes that can occur as some such $A[p]$. In this paper, we study which group schemes can occur as $J[p]$, where $J$ is the Jacobian of a hyperelliptic curve defined over $\mathbb{F}_p$. We do this by computing explicit formulae for the action of Frobenius and its dual on the de Rham cohomology of a hyperelliptic curve with respect to a given ba…
In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.
Let $\ell$ be a rational prime and $k$ a number field. Given a superelliptic curve $C/k$ of $\ell$-power degree, we describe the field generated by the $\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction type of $C$ (and hence, in terms of data determined by a suitable affine model of $C$). If the Jacobian is good away from $\ell$ and the branch set is defined over a pro-$\ell$ extension of $k(μ_{\ell^\infty})$ unramified away from $\ell$, then the $\ell$-power torsion of the Jacobian is rational over the maximal such extension. By decomposing the covering int…
We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion of an `almost rational torsion point.'
In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with "ordinary good" or "ordinary semistable" reduction at a given prime. We also give new proofs of: (1) The Manin-Mumford conjecture: There are only finitely many torsion points lying on a curve of genus at least 2 embedded in its Jacobian by an Albanese map; and (2) The Coleman-Kaskel-Ribet conjecture: If p is a prime number which is at least 23, then the only torsion po…
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…
Let $k$ be a field of characteristic $0$, and let $α_{1}$, $α_{2}$, ..., $α_{5}$ be algebraically independent and transcendental over $k$. Let $K$ be the transcendental extension of $k$ obtained by adjoining the elementary symmetric functions of the $α_{i}$'s. Let $J$ be the Jacobian of the hyperelliptic curve defined over $K$ which is given by the equation $y^{2} = \prod_{i = 1}^{5} (x - α_{i})$. We define a tower of field extensions $K = K_{0}' \subset K_{1}' \subset K_{2}' \subset ...$ by giving recursive formulas for the generators of each $K_{n}'$ over $K_{n - 1}'$, and let $K_{\infty}' =…
We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves.
We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Pic…
We consider the generalized Jacobian $\widetilde{J}$ of the modular curve $X_0(N)$ of level $N$ with respect to a reduced divisor consisting of all cusps. Supposing $N$ is square free, we explicitly determine the structure of the $\mathbb{Q}$-rational torsion points on $\widetilde{J}$ up to $6$-primary torsion. The result turns out to be very different from the case of prime power level previously studied by Yang and the second author. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first …
By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it is known that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve of discriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-new subvariety of J_0^D(pqN), whenever p and q are distinct primes not dividing D or N. However, this approach is entirely nonconstructive, and gives no information at all about any particular isogeny between the two varieties. In this paper, we determine the kernels of all possible maps from one variety to the other up to support on a small finite set of maximal idea…
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…
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.
Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial self-pairings of the $\ell$-Tate pairing in terms of the action of the Frobenius on the $\ell$-torsion of the Jacobian of a genus 2 curve. We apply similar techniques to study the non-degeneracy of the $\ell$-Tate pairing restrained to subgroups of the $\ell$-torsion which are maximal isotropic with respect to the Weil pairing. First, we deduce a criterion to verify whether the jacobian of a genus 2 curve has maximal endomorphism ring. Secondly, we derive a method to construct horizontal $(\ell,\ell)$-isogenies starting …
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 $A_f$.
In this paper we present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.
Consider the jacobian of a hyperelliptic genus two curve defined over a finite field. Under certain restrictions on the endomorphism ring of the jacobian we give an explicit description all non-degenerate, bilinear, anti-symmetric and Galois-invariant pairings on the jacobian. From this description it follows that no such pairing can be computed more efficiently than the Weil pairing. To establish this result, we need an explicit description of the representation of the Frobenius endomorphism on the l-torsion subgroup of the jacobian. This description is given. In particular, we show that if t…
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 …
We address the problem of computing in the group of $\ell^k$-torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations.
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…
Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. Our main result is that every $BT_1$ scheme group over $k$ occurs as a direct factor of the $p$-torsion group scheme of the Jacobian of an explicit curve defined over $\mathbb{F}_p$. We also treat a variant with polarizations. Our main tools are the Kraft classification of $BT_1$ group schemes, a theorem of Oda, and a combinatorial description of the de Rham …
