Algebraic approaches to the Brauer-Siegel ratio for abelian varieties over function fields
Series: Algebra and Number Theory Seminar
Location: ENR2 S395
Presenter: Doug Ulmer, U of A
In analogy with the classical Brauer-Siegel theorem, Marc Hindry proposed the study of the ratio log ( (order of sha)(Neron-Tate regulator) ) / log( exponential differential height ) for families of abelian varieties over a fixed global field. Later, Hindry-Pacheco and Griffon proved a number of interesting results about this ratio, including a computation of the limit for several natural families, over global function fields of characteristic p. Their arguments use the formula of Birch and Swinnerton-Dyer for the leading term of the L-function of the abelian variety and analytic arguments. After explaining the motivation and background, we will explain how to recover (and extend) some of their results by algebraic means, namely by a direct analysis of the Tate-Shafarevich group.