Jeroen Sijsling

ON COMPUTING BELYI MAPS

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.

FAST COMPUTATION OF ISOMORPHISMS OF HYPERELLIPTIC CURVES AND EXPLICIT GALOIS DESCENT

We show how to speed up the computation of isomorphisms of hyperelliptic curves by using covariants. We also obtain new theoret- ical and practical results concerning models of these curves over their eld of moduli.

An explicit expression of the Lüroth invariant

In this short note, we give an algorithm that returns an explicit expression of the Lüroth invariant in terms of the Dixmier-Ohno invariants of plane quartic curves. We also obtain an explicit factorized expression on the locus of Ciani quartics in terms of the coefficients. After this calculation, we extend our methods to answer two open theoretical questions concerning the sub-locus of singular Lüroth quartics.

A database of genus-2 curves over the rational numbers

We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. This data has been incorporated into the L-Functions and Modular Forms Database (LMFDB).

Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

This paper is devoted to the study of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism groups are cyclic of order coprime to the characteristic of their ground field. We give an explicit and effectively computable description of this obstruction. Along the way, we obtain an arithmetic criterion for the existence of a so-called hyperelliptic descent. We define homogeneous dihedral invariants for general hyperelliptic curves, and show how the obstruction can be expressed in terms of these invariants. If this obstruction vanishes, then the homogeneous dihedral invariants can also be used to explicitly construct a model over the field of moduli of the curve; if not, then one still obtains a hyperelliptic model over a degree 2 extension of the field of moduli.

Rigorous computation of the endomorphism ring of a Jacobian

We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.

On explicit descent of marked curves and maps

We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to Dèbes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.

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.

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ABSTRACT We analyze the variation around the mean of the distribution of the number of rational points on non-hyperelliptic genus 3 curves over finite fields, by extrapolating from results on the distribution of traces of Frobenius for plane curves whose degree is small with respect to the cardinality of their finite base field. We put our results in perspective with a numerical study for prime fields of characteristic 11 ⩽ p ⩽ 53. Our methods shed some new light on the asymmetry of the distribution around its mean value, which is related to the Serre obstruction.