Does hyperbolic 3-geometry provide an infinite family of fields with class number one?
Friday, 21.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The class number one problem dating back to Gauss' work on quadratic\nforms asks whether infinitely many number fields of ideal class number\none exist. In an exciting 2017 paper, Ulf Rehmann and Ernest Vinberg\nhave described a candidate family of fields defined in hyperbolic\n3-geometry which, in all computed examples, turned out to have class\nnumber one. In the talk, we will introduce this family of fields and\nexamine its class number phenomenon from different perspectives: by an\nanalogy to a known class number formula in hyperbolic 3-geometry, by\nempirical computations and by estimates with known class number\nstatistics.
The standard conjecture of Hodge type for abelian fourfolds
Friday, 28.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The standard conjecture of Hodge type for abelian fourfolds
Friday, 28.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Let S be a surface, V be the Q-vector space of divisors on S modulo numerical equivalence and d be the dimension of V. The intersection product defines a non degenerate quadratic form on V. The Hodge index theorem says that it is of signature (1,d-1).\nIn the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is a consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable, thanks to p-adic Hodge theory. Moreover, using classical product formulas on quadratic forms, the p-adic result will give non-trivial information on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.