Primes dividing invariants of CM Picard curves

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof, and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves

Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

We study a 3-dimensional stratum ℳ3 , V of the moduli space ℳ3 of curves of genus 3 parameterizing curves Y that admit a certain action of V = C2 × C2. We determine the possible types of the stable reduction of these curves to characteristic different from 2. We define invariants for ℳ3 , V and characterize the occurrence of each of the reduction types in terms of them. We also calculate the j-invariant (respectively the Igusa invariants) of the irreducible components of positive genus of the stable reduction Y in terms of the invariants.</p

Modular invariants for genus 3 hyperelliptic curves

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM

A bound on the primes of bad reduction for CM curves of genus 3

16 pages, some minor and major updatesWe give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are essential for algorithmic construction of curves with given characteristic polynomials over finite fields

Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

We study a 3-dimensional stratum ℳ3 , V of the moduli space ℳ3 of curves of genus 3 parameterizing curves Y that admit a certain action of V = C2 × C2. We determine the possible types of the stable reduction of these curves to characteristic different from 2. We define invariants for ℳ3 , V and characterize the occurrence of each of the reduction types in terms of them. We also calculate the j-invariant (respectively the Igusa invariants) of the irreducible components of positive genus of the stable reduction Y in terms of the invariants