Adelina Manzateanu

Invariants of hyperelliptic genus 3 curves

A curve of genus 3 is either hyperelliptic or a plane quartic. When considered separately, the hyperelliptic and the plane quartic loci, each have a standard set of invariants. For many applications, such as the explicit construction of genus 3 curves with complex multiplication (CM), it would be interesting to relate these invariants to a set of modular analogues, that can be computed in terms of a generating set of Siegel modular functions. In joint work with Sorina Ionica, Pinar Kilicer, Kristin Lauter, Elisa Lorenzo Garcia, Maike Massierer and Christelle Vincent, we introduce some modular functions of degree 3 which, when evaluated on the hyperelliptic locus, yield values whose denominator contain the primes of bad reduction for the corresponding hyperelliptic curve.
In this talk, I will use the complete set of hyperelliptic curves of genus 3 with CM defined over Q to illustrate our result.
Moreover, I will explain how this relates to the embedding problem as posed by Goren and Lauter.