A Simple Method for Obtaining Relations Among Factor Basis Elements for Special Hyperelliptic Curves

Nagao had proposed a decomposition method for divisors of hyperelliptic curves defined over a field \rF_{q^n} with $n\geq 2$. Joux and Vitse had later proposed a variant which provided relations among the factor basis elements. Both Nagao\u27s and the Joux-Vitse methods require solving a multi-variate system of non-linear equations. In this work, we revisit Nagao\u27s approach with the idea of avoiding the requirement of solving a multi-variate system. While this cannot be done in general, we are able to identify special cases for which this is indeed possible. Our main result is for curves $C:y^2=f(x)$ of genus $g$ defined over \rF_{q^2} having characteristic greater than two. If $f(x)$ has at most $g$ consecutive coefficients which are in \rF_{q^2} while the rest are in \rF_q, then we show that it is possible to obtain a single relation in about $(2g+3)!$ trials. The method combines well with a sieving method proposed by Joux and Vitse. Our implementation of the resulting algorithm provides examples of factor basis relations for $g=5$ and $g=6$. We believe that none of the other methods known in the literature can provide such relations faster than our method. Other than obtaining such decompositions, we also explore the applicability of our approach for $n>2$ and also for binary characteristic fields

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t = −1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added

Thomae type formulae for singular Z_N curves

We give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs.Comment: 22 page

Special subvarieties arising from families of cyclic covers of the projective line

We consider families of cyclic covers of the projective line, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques in mixed characteristics due to Dwork and Ogus.Comment: Minor improvements. To appear in Documenta Mat

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 D\`ebes 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.Comment: 35 page