Sep 09  Daniel LiHuerta (Harvard)

The Plectic conjecture over local fields
The étale cohomology of varieties over Q enjoys a Galois action. In the case of Hilbert modular varieties, NekovářScholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. Motivated by applications to higherrank Euler systems, they conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.
We present a proof of the analog of this conjecture for local Shimura varieties. Consequently, we obtain results for the basic locus of global Shimura varieties, after restricting to a decomposition group. The proof crucially uses a mixedcharacteristic version of fusion due to FarguesScholze.

Sep 16 (Online)  Jinbo Ren (Xiamen University)

Applications of the Subspace Theorem in Group Theory
An abstract group is said to have the bounded generation property (BG) if it can be written as a product of finitely many cyclic subgroups. Being a purely combinatorial notion, bounded generation has close relation with many group theoretical problems including semisimple rigidity, Kazhdan's property (T) and Serre's congruence subgroup problem.
In this talk, I will explain how to use the SchlickeweiSchmidt subspace theorem in Diophantine approximation to prove that the distribution of the points of a set of matrices over a number field $F$ with (BG) by certain fixed semisimple (diagonalizable) elements is of at most logarithmic size when ordered by height. Moreover, one obtains that a linear group $\Gamma \subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus.
This is joint work with Corvaja, Demeio, Rapinchuk and Zannier.

Sep 23  Oana Padurariu (Boston)

On Quadratic Analogues of Kenku's Theorem
Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$, building on Mazur's 1978 work on prime degree isogenies. Although more than
40 years have passed, the determination of cyclic isogenies of elliptic curves over a single
other number field has until now not been realized.
In this talk I will present a procedure
to assist in establishing such a determination for a given quadratic field. Running this
procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $\lvert d \lvert < 10^4$ we obtain, conditional on the
GRH, the determination of cyclic isogenies of elliptic curves over 19 quadratic fields.
This is joint work with Barinder Banwait and Filip Najman.

Sep 30  Chandrashekhar Khare (UCLA)

The WilesLenstraDiamond numerical criterion in higher codimensions
I will report on recent joint work with Srikanth Iyengar and Jeff Manning. We give a development of numerical criterion that was used by Wiles as an essential ingredient in
his approach to modularity of elliptic curves over $\mathbb{Q}$. The patching method introduced by Wiles and Taylor has been developed considerably while the numerical criterion has lagged behind.
We prove new commutative algebra results that lead to a generalisation of the WilesLenstraDiamond numerical criterion in situations of positive defect (as arise when proving modularity of elliptic curves over number fields with a complex place). A key step in our work is the definition of congruence modules in higher codimensions which should be relevant to studying properties of eigenvarieties at classical points.

Oct 07  Naomi Sweeting (Harvard)


Oct 14  Andrew Obus (BaruchCUNY)


Oct 21  Jacques Tilouine (Sorbonne Paris Nord)


Oct 28 


Nov 04  Hélène Esnault (Freie Universität Berlin)


Nov 11  Eric Chen (Princeton)


Nov 18  Robert Pollack (Boston)


Dec 02 


Dec 09  Ari Shnidman (HUJ & Dartmouth)

