special seminar

Integral points and curves on moduli of local systems

Junho Peter Whang
Princeton University
December 8, 2017
The classical affine cubic surface of Markoff has a well-known interpretation as a moduli space for local systems on the once-punctured torus. We show that the analogous moduli spaces for general topological surfaces form a rich family of log Calabi-Yau varieties, where a structure theorem for their integral points can be established using mapping class group descent. Related analysis also yields new results on the arithmetic of algebraic curves in these moduli spaces, including finiteness of imaginary quadratic integral points for non-special curves.

Integral points on Markoff-type cubic surfaces

Amit Ghosh
Oklahoma State University
December 8, 2017
We report on some recent work with Peter Sarnak. For integers $k$, we consider the affine cubic surfaces $V_k$ given by $M(x) = x_1^2 + x_2 + x_3^2 − x_1 x_2 x_3 = k$. Then for almost all $k$, the Hasse Principle holds, namely that $V_k(Z)$ is non-empty if $V_k(Z_p)$ is non-empty for all primes $p$. Moreover there are infinitely many $k$'s for which it fails. There is an action of a non-linear group on the integral points, producing finitely many orbits. For most $k$, we obtain an exact description of these orbits, the number of which we call "class numbers".

Diophantine analysis in thin orbits

Alex Kontorovich
Rutgers University; von Neumann Fellow, School of Mathematics
December 8, 2017
We will explain how the circle method can be used in the setting of thin orbits, by sketching the proof (joint with Bourgain) of the asymptotic local-global principle for Apollonian circle packings. We will mention extensions of this method due to Zhang and Fuchs-Stange-Zhang to certain crystallographic circle packings, as well as the method's limitations.

How to modify the Langlands' dual group

Joseph Bernstein
Tel Aviv University; Member, School of Mathematics
November 20, 2017

Let $\mathcal G$ be a split reductive group over a $p$-adic field $F$, and $G$ the group of its $F$-points.

The main insight of the local Langlands program is that to every irreducible smooth representation $(\rho, G, V )$ should correspond a morphism $\nu_\rho : W_F \to {}^\vee G$ of the Weil group $W_F$ of the field $F$ to the Langlands' dual group $^\vee G$.

Wild harmonic bundles and related topics II

Takuro Mochizuki
Kyoto University
November 17, 2017

Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a projective manifold has a pluri-harmonic metric. Moreover, Simpson generalized many important theorems for polarizable variation of Hodge structures, such as Hard Lefschetz Theorem, to the context of harmonic bundles.

Wild harmonic bundles and related topics I

Takuro Mochizuki
Kyoto University
November 15, 2017

Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a projective manifold has a pluri-harmonic metric. Moreover, Simpson generalized many important theorems for polarizable variation of Hodge structures, such as Hard Lefschetz Theorem, to the context of harmonic bundles.

Modulo $p$ representations of reductive $p$-adic groups: functorial properties

Marie-France Vignéras
Institut de Mathématiques de Jussieu
November 30, 2016
Let $F$ be a local field with finite residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf G$ be a connected reductive $F$-group. With Abe, Henniart, Herzig, we classified irreducible admissible $C$-representations of $G=\mathbf G(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. For a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $\tau$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G \tau$.