A Brauer-Siegel theorem for an Artin-Schreier family of elliptic curves

The classical Brauer-Siegel theorem gives upper and lower bounds on the product of the class-number times the regulator of units of a number field, in terms of its discriminant. Now consider an elliptic curve E defined over $F_q (t)$: one can form the product of the order of the Tate-Shafarevich group of E (assuming it is finite) and of its Néron-Tate regulator. We are interested in finding upper and lower bounds of this quantity in terms of simpler invariants of E, e.g. its height. In general this question is open, and has a satisfactory answer in only a handful of cases. In this talk, I will report on a recent work where I studied an "Artin-Schreier family" of elliptic curves. I will explain how good unconditional bounds can be found in this case. This provides a new example of family of elliptic curves for which an analogue of the classical Brauer-Siegel theorem holds. T.B.A.