Weights in arithmetic geometry

Arithmetic in Geometry

I thank Dean Martin for his very flattering introduction. I thank the College for bestowing a chair to me and the Mathematics Department for providing a wonderful atmosphere for my work. I thank Don Lewis and my current and former colleagues for their support. I take this opportunity to also thank my family, friends, and collaborators. My mathematical horizon has widened considerably after join...

Galois representations in arithmetic geometry

Takeshi SAITO When he formulated an analogue of the Riemann hypothesis for congruence zeta functions of varieties over finite fields, Weil predicted that a reasonable cohomology theory should lead us to a proof of the Weil conjecture. The dream was realized when Grothendieck defined etale cohomology. Since then, -adic etale cohomology has been a fundamental object in arithmetic geometry. It ena...

Polylogarithms in Arithmetic and Geometry

After a century of neglect the dilogarithm appeared twenty years ago in works of Gabrielov-Gelfand-Losik [GGL] on a combinatorial formula for the first Pontryagin class, Bloch on K-theory and regulators [Bl1] and Wigner on Lie groups. The dilogarithm has a single-valued cousin : the Bloch Wigner function L2(z) := ImLi2(z) + arg(1− z) log |z|. Let r(x1, ..., x4) be the cross-ratio of 4 distinct ...

Arithmetic Algebraic Geometry

[3] , Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Finiteness results for modular curves of genus at least 2, Amer.

Arithmetic and Diophantine Geometry