Every smooth proper algebraic variety over a p-adic field is expected to have a semistable model after passing to a finite extension. This conjecture is open in general, but its analogue for Galois representations, the p-adic monodromy theorem, is known. In this talk, we will explain a generalization of this theorem to étale local systems on a smooth rigid analytic variety.
We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some interesting relations between shtuka spaces and function field arithmetic. For instance, there should be a notion of modularity for the r-fold product of an elliptic curve. We verify these predictions in a few cases.
This is partly joint work with Noam Elkies.
We will discuss the problem of constructing and characterizing uniquely, integral models of Shimura varieties over some primes where non-smooth reduction is expected.
The singularities in the reduction modulo p of the modular curve Y0(p) are visualized by the famous picture of two curves meeting transversally at the supersingular points. It is a fundamental question to understand the singularities which arise in the reductions modulo p of integral models of Shimura varieties. For PEL type Shimura varieties with parahoric level structure at p, this question has been studied since the 1990s. Due to the recent construction of Kisin and Pappas, it now makes sense to pursue this question for abelian type Shimura varieties with parahoric level structure. Recently He-Pappas-Rapoport gave a classification of the Shimura varieties in this class which have either good or semistable reduction. But what is the strongest statement we can make about the nature of the singularities in general? For some time it has been expected that the integral models are Cohen-Macaulay.
This talk will discuss recent work with Timo Richarz, in which we prove that, with mild restrictions on p, all Pappas-Zhu parahoric local models, and therefore all Kisin-Pappas Shimura varieties, are Cohen-Macaulay.
On an elliptic curve y2=x3+ax+b, the points with coordinates (x,y) in a given number field form a finitely generated abelian group. One natural question is how the rank of this group changes when changing the number field.
For the simplest example with infinitely many number fields, fix a prime p. Adjoining to ℚ the pth, p2th, p3th,... roots of unity produces a *tower* of number fields
ℚ⊂ℚ(ζp)⊂ℚ(ζp2)⊂....
One may guess that the rank should keep growing in this tower ('more numbers mean more solutions'). However, this guess turns out to be incorrect: the rank is always bounded, as envisioned by the theories of Iwasawa and Mazur in the 1970s.
The above tower started with ℚ, but there are analogous towers that start with an imaginary quadratic field instead. Given the above boundedness result, one would now guess that the rank is bounded in these towers, too. Surprisingly, this is not the case: there are scenarios both for bounded and unbounded rank. So how does the rank grow in those towers in general? We initiate an answer to this question in this talk. This is joint work with Antonio Lei.
The seminal work of Deligne and Lusztig on the representations of finite reductive groups has influenced an industry studying parallel constructions in the same theme. In this talk, we will discuss recent progress on studying analogues of Deligne-Lusztig varieties attached to p-adic groups.
Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) where D' is a central division K-algebra of degree n having the same isomorphism classes of maximal subfields as D. I will review the results on gen(D) obtained in the last several years, in particular the finiteness theorem for gen(D) when K is finitely generated of characteristic not dividing n. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic K-groups using maximal K-tori in place of maximal subfields, and report on some recent progress in this direction. (Joint work with V. Chernousov and I. Rapinchuk)
