Simone Muselli

Models and Integral Differentials of Hyperelliptic Curves

Let C: y^2=f(x) be a hyperelliptic curve defined over the rationals. A recent paper by T. Dokchitser, V. Dokchitser, C. Maistret, A. Morgan introduces a new combinatorial tool, the cluster picture, that measures the p-adic distance between the roots of f. It turns out that many arithmetic objects associated to C can be computed from cluster picture invariants. In my talk I will present my latest work in which I used this setting (and some new definitions) to compute a regular model and a basis of integral differentials of C explicitly. In particular, it gives an explicit formula to compute one of the invariants in the Birch and Swinnerton-Dyer conjecture for (the Jacobian of) C.