arXiv subjects
Publications and source records attributed to Claus Diem.
For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.
We show that for all odd primes $p$, there exist ordinary elliptic curves over $\bar{\mathbb{F}}_p(x)$ with arbitrarily high rank and constant $j$-invariant. This shows in particular that there are elliptic curves with arbitrarily high rank over these fields for which the corresponding elliptic surface is not supersingular. The result follows from a theorem which states that for all odd prime numbers $p$ and $\ell$, there exists a hyperelliptic curve over $\bar{\mathbb{F}}_p$ of genus $(\ell-1)/2$ whose Jacobian is isogenous to the power of one ordinary elliptic curve.
We present an index calculus algorithm with double large prime variation which lends itself well to a rigorous analysis. Using this algorithm we prove that for fixed genus $g \geq 2$, the discrete logarithm problem in degree 0 class groups of non-singular curves over finite fields $\mathbb{F}_q$ can be solved in an expected time of $\tilde{O}(q^{2-2/g})$, provided that the curve is given by a plane model of bounded degree and the degree 0 class group is cyclic. The result generalizes a previous result for hyperelliptic curves given by an imaginary Weierstraß equation obtained by Gaudry, Thomé, Thériault and the author.
We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism). A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic different from 2 is birational to a product of two elliptic curves over the projective line. We formulate and prove a generalization of this theorem for the relative situation. We also prove a Torelli theorem for genus 2 curves over an arbitrary base.
We give a description of endomorphism rings of Weil restrictions of abelian varieties with respect to finite Galois extensions. The results are applied to study the isogeny decomposition of Weil restrictions.
We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a theorem which states that whenever p is a generator of (Z/ell Z)^*/<-1> (ell an odd prime) there exists a hyperelliptic curve over $\bar F_p$ whose Jacobian is isogenous to a power of one ordinary elliptic curve.