Marco Streng

Computing Igusa class polynomials

We give an algorithm that computes the genus-two class polynomials of a primitive quartic CM-eld K, and we give a running time bound and a proof of correctness of this algorithm. This is the rst proof of correctness and the rst running time bound of any algorithm that computes these polynomials. Our algorithm is based on the complex analytic method of Spallek and van Wamelen and runs in time e O( 7=2 ), where is the discriminant of K.

Abelian varieties with prescribed embedding degree

We present an algorithm that, on input of a CM-field K, aninteger k ≥ 1, and a prime r ≡ 1 mod k, constructs a q-Weil numberπ ∈ OK corresponding to an ordinary, simple abelian variety A overthe field F of q elements that has an F-rational point of order r andembedding degree k with respect to r. We then discuss how CM-methods over K can be used to explicitly construct A.

A CM construction for curves of genus 2 with p-rank 1

We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field Fp2 of p2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of Fp2-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over Fp2 out of necessity: we show that curves of p-rank 1 over Fp for large p cannot be efficiently constructed using explicit CM constructions.

Algebraic Divisibility Sequences Over Function Fields

In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

On polarised class groups of orders in quartic CM-fields

We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing endomorphism rings of abelian surfaces over finite fields, and we use our results to extend a completeness result of Murabayashi and Umegaki to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.

Plane quartics over

We give examples of smooth plane quartics over QQQ with complex multiplication over Q¯¯¯¯Q¯\overline{Q} by a maximal order with primitive CM type. We describe the required algorithms as we go, these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and reconstruction from these invariants. Finally, we discuss some of the reduction properties of the curves that we obtain.