Marc Hindry

Analogues of Brauer-Siegel theorem in arithmetic geometry

We will explain analogies between the classical Brauer-Siegel theorem, a statement relating asymptotically the class number, regulator of units and discriminant of a number field, and similar statement involving arithmetic invariants of algebraic varieties over a finite or global field. We present precisely the analogy for surfaces over a finite field and for abelian varieties over a global field (i.e. a number field or the function field of a cuve over a finite field), surveying some recent results (our own and Griffon, Ulmer). The proof of Brauer-Siegel theorem relies on the class number formula and analytical estimates for the Dedekind zeta function, the analogy draws on formulae predicted by the Birch & Swinnerton-Dyer conjecture, (resp. Artin-Tate conjecture) and analytical estimates for the relevant L-series. If time permits, we will also formulate a quite general question along these lines, for algebraic varieties over a global field, and develop the case of projective hypersurfaces.