F. Oort showed that the moduli space of principally polarized supersingular abelian surfaces is a union of rational curves. This is proven by showing that every principally polarized supersingular abelian surface is the jacobian of a fibre of one of the families of genus 2 curves $\pi: \mathcal{C}\rightarrow \mathbb{P}^1$ constructed by L. Moret-Bailly. We present an algorithm that makes this construction effective: Given a point $x\in \mathbb{P}^1$ we compute a hyperelliptic model of the fibre $\pi^{-1}(x)$. The algorithm uses Mumford's theory of theta groups to compute
quotients by the group scheme $\alpha_p$.
The talk is based on a paper recently submitted for publication by G.J. vd Heiden, J. Top and myself, which in turn is based on my masters' thesis (which in turn is based on vd Heidens masters' thesis...).
We consider for a prime p>3 the hyperelliptic curve C defined over the rationals by the equation y²=x(x²-p²)(x²-4p²). The talk can be divided into three parts:
- Rédei symbols, which satisfy a reciprocity law based on the product formula for quadratic Hilbert symbols in quadratic number fields.
- Rankbounds on J = Jac(C) over Q obtained by a 2-Selmergroup computation, which for certain classes of primes p of positive Dirichlet density can be improved by the use of Rédei symbols.
- Determination of the rational points of C for some primes p by use of 'baby-Elliptic Curve Chabauty' (baby because in our case, which is working over Q, we can completely avoid the general machinery of the method). For example, I will prove that for p = 5 mod 24 (even though J(Q) has rank 1 if Sha(J/Q) is finite), the rational points C(Q) consist of the Weierstrass points, unless p=5.
In the first part of the talk, I will give a gentle introduction into the subject of divisibility sequences over the
rational numbers and discuss the notion of a primitive divisor/Zsigmondy bound. I then explain how these notions can be extended to number fields and function fields, and how to obtain a divisibility sequence from a non-torsion point on an elliptic curve over any of these fields. I will also provide some nice examples.
In the second part of the talk, we discuss the typical methods that are used to prove the existence of a Zsigmondy bound for a divisibility sequence obtained from a non-torsion point on an elliptic curve E over a function field K (but the same approach works for number fields). Let P be this non-torsion point in E(K), and suppose Q is a torsion point in E(K). We can also associate a sequence of divisors {D_{nP+Q}} on K to the sequence of points {nP+Q}.
In my preprint, I have proved the existence of a Zsigmondy bound for this sequence {D_{nP+Q}} (under some minor conditions), extending the analogous result of Verzobio over number fields. I end my talk by providing the crucial ideas I used to be able to apply the existing methods to my case {D_{nP+Q}}.
Jacobians of CM curves are abelian varieties with a particularly large endomorphism algebra, which provides them with a rich arithmetic structure. The motivating question for the results in this talk is whether we can find hyperelliptic and non-hyperelliptic curves with maximal CM by a given order whose Jacobians are isogenous.
Joint work with Sorina Ionica considers this question in genus 3 by usingthe catalogue of CM fields in the LMFDB, and found a (small) list of such isogenous Jacobians. This talk describes the main constructions and results. Furthermore, it gives a classification of CM types and their reflex fields for this particular genus.
Resuming the theme of last week, we consider the Galois action on the set of Jacobians with given CM, and use the Shimura class group to get some grip on the size of the corresponding orbits. This leads to algorithms that given a maximal CM order ZZ_K determine a small set of period matrices that represent all these Galois orbits. Using them and evaluating theta values by work of Labrande gives the results mentioned in the first talk. We also discuss how (heuristically) to determine defining equations for the corresponding CM curves.
I will report on my long term efford to make the computation of the semistable reduction of curves over p-adic fields effective and practical. I will focus on some particular cases, and on the use of methods from nonarchimedian analytic geometry to achieve this goal.
