Related papers: Computing isogenies from modular equations in genu…

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class…

We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite…

Let $Y$ be a genus $2$ curve over $\mathbb Q$. We provide a method to systematically search for possible candidates of a prime $\ell\geq 3$ and a genus $1$ curve $X$ for which there exists a genus $3$ curve $Z$ over $\mathbb Q$ whose…

We determine the possible degrees of cyclic isogenies defined over quadratic fields for non-CM elliptic curves with rational $j$-invariant.

Given a genus two curve $X: y^2 = x^5 + a x^3 + b x^2 + c x + d$, we give an explicit parametrization of all other such curves $Y$ with a specified symplectic isomorphism on three-torsion of Jacobians $\mbox{Jac}(X)[3] \cong…

Let $A/\overline{\mathbb{F}}\_p$ and $A'/\overline{\mathbb{F}}\_p$ be supersingular principally polarized abelian varieties of dimension $g>1$. For any prime $\ell \ne p$, we give an algorithm that finds a path $\phi \colon A \rightarrow…

In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show…

Let $\mathcal X$ be a genus 2 curve defined over a field $K$, $\mbox{char} K = p \geq 0$, and $\mbox{Jac} (\mathcal X, \iota)$ its Jacobian, where $\iota$ is the principal polarization of $\mbox{Jac} (\mathcal X)$ attached to $\mathcal X$.…

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component…

Suppose $C$ is an isogeny class of abelian varieties over a finite field $k$. In this paper we give a partial answer to the question of which finite group schemes over $k$ occur as kernels of polarizations of varieties in $C$. We show that…

For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally non-equivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the…

The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose…

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…

We develop a method to compute the Ekedahl-Oort type of a curve C over a field k of characteristic p (which is the isomorphism type of the p-kernel group scheme J[p], where J is the Jacobian of C). Part of our method is general, in that we…

Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree…

We compute equations for real multiplication on the divisor classes of genus two curves via algebraic correspondences. We do so by implementing van Wamelen's method for computing equations for endomorphisms of Jacobians on examples drawn…

We compute the intersections between the automorphism strata and the pullback by the Torelli map of the Ekedahl-Oort strata inside the moduli space of genus two curves. We first describe explicitly which possible automorphism groups a genus…

It is well known that the Prym variety of an \'etale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny…

Isogenies occur throughout the theory of elliptic curves. Recently, the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols. Given two elliptic curves $E_1, E_2$ defined over…

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…