L6

In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be p-adically interpolated. In other words, Eisenstein series can be viewed as specializations of a p-adic family parametrized by the weight. The notion of p-adic variations of modular forms was later generalized by Hida to include families of ordinary cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic space classifying much more general p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight k ≥ 2, while the weight one case is far more intricate.

In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. Via this Galois-theoretic method, we obtain the q-expansion of some non-classical overconvergent forms in terms of p-adic logarithms of p-units in certain number field. Finally, we will explain how these calculations suggest a different approach to the Gross-Stark conjecture.

I will give an overview of some recent progress on potential automorphy results over CM fields, that is joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne. I will focus on explaining an application to the generalized Ramanujan-Petersson conjecture. 

For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down down to -X for which the class group has trivial (non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss recent refinements of these classic results in which we consider the imaginary quadratic fields whose class number is indivisible (divisible) by p such that a given finite set of primes factor in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups satisfying almost any given finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

Let F be a totally real or CM number field.  Scholze has constructed Galois representations associated with torsion classes in the cohomology of locally symmetric spaces for GL_n(F).  We show that the nilpotent ideal appearing in Scholze's construction can be removed when F splits completely at the relevant prime.  As a key component of the proof, we show that the compactly supported cohomology of certain unitary and symplectic Shimura varieties with level  Gamma_1(p^{\infty}) vanishes above the middle degree. This is joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih. 

Understanding how shifts of multiplicative functions correlate with each other is a central question in multiplicative number theory. A well-known conjecture of Elliott predicts that there should be no correlation between shifted multiplicative functions unless the functions involved are ‘pretentious functions’ in a certain precise sense. The Elliott conjecture implies as a special case the famous Chowla conjecture on shifted products of the Möbius function.

In the last few years, there has been a lot of exciting progress on the Chowla and Elliott conjectures, and we give an overview of this. Nearly all of the previously obtained results have concerned correlations that are weighted logarithmically, and it is an interesting question whether one can remove these logarithmic weights. We show that one can indeed remove logarithmic averaging from the known results on the Chowla and Elliott conjectures, provided that one restricts to almost all scales in a suitable sense.

This is joint work with Terry Tao.

If k is a field of characteristic zero, a theorem of Lurie and Pridham establishes an equivalence between formal moduli problems and differential graded Lie algebras over k. We generalise this equivalence in two different ways to arbitrary ground fields by using “partition Lie algebras”. These mysterious new gadgets are intimately related to the genuine equivariant topology of the partition complex, which allows us to access the operations acting on their homotopy groups (relying on earlier work of Dyer-Lashof, Priddy, Goerss, and Arone-B.). This is joint work with Mathew.

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $\eta$. In recent joint work with Tim Gowers, we were aiming to obtain a qualitative description of closed subsets of the Cayley graph $\Gamma$ whose vertex set is $\mathbb{F}_2^{n_1}\otimes \dots \otimes \mathbb{F}_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdots \otimes u_d$. For the matrix case (that is, when $d=2$), such a description was obtained by Khot, Minzer and Safra, a breakthrough that completed the proof of the 2-to-2 conjecture. We have formulated a conjecture for higher dimensions, and proved it in an important special case. In this talk, I will sketch this proof. Also, we have identified a statement about $\eta$-closed sets in Cayley graphs on arbitrary finite Abelian groups that implies the conjecture and can be considered as a "highly asymmetric Balog-Szemerédi-Gowers theorem" when it holds. I will present an example to show that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph $\Gamma$.

In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. For each d we prove the hyperbolicity of all but (perhaps) finitely many Jensen polynomials. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This result can be thought of as a proof of GUE for the Riemann zeta function in derivative aspect. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.
 

We will see some results and conjectures on the zeta and multizeta values in the function field context, and see how they relate to homological-homotopical objects, such as t-motives, iterated extensions, and to Hopf algebras, big Galois representations.

In directed algebraic topology, a topological space is endowed 
with an extra structure, a selected subset of the paths called the 
directed paths or the d-structure. The subset has to contain the 
constant paths, be closed under concatenation and non-decreasing 
reparametrization. A space with a d-structure is a d-space.
If the space has a partial order, the paths increasing wrt. that order 
form a d-structure, but the circle with counter clockwise paths as the 
d-structure is a prominent example without an underlying partial order.
Dipaths are dihomotopic if there is a one-parameter family of directed 
paths connecting them. Since in general dipaths do not have inverses, 
instead of fundamental groups (or groupoids), there is a fundamental 
category. So already at this stage, the algebra is less desirable than 
for topological spaces.
We will give examples of what is currently known in the area, the kind 
of methods used and the problems and questions which need answering - in 
particular with applications in computer science in mind.