Schedule

All talks are located in Lecture Hall -101 of the Old University building at Rheinsprung 9 in Basel. Link to OpenStreetMap

Friday, November 10

TimeSpeakerTitleAbstract
14.00 - 15.00 Javier Fresán Fixed-point statistics from spectral measures I will report on a project with Arthur Forey and Emmanuel Kowalski that grew out of some afterthoughts on our work on equidistribution of exponential sums. We define spectral measures associated with complex-valued additive invariants on tensor categories, and find simple criteria for their existence and uniqueness. We then compute them for some exotic tensor categories, such as Deligne's category of representations of the ``symmetric group'' $S_t$ for a complex number $t$, and show how they give rise to abstract proofs of old and new fixed-point statistics results, such as the fact that the random variables giving the number of fixed points of a uniformly chosen random permutation on $n$ letters converge to the Poisson distribution with parameter $1$ as $n$ goes to infinity. Since these are the Number Theory Days, I won't resist speculating about a generalisation of Chebotarev's density theorem to certain pseudopolynomials, with $S_t$ playing the role of the symmetric group$\ldots$
15.00 - 15.30
Coffee and tea break
15.30 - 16.30 Vesselin Dimitrov No-shadowing bounds for polynomials and L-functions An old idea in analytic number theory invites a comparison of the zeros of a suitably irreducible L-function (such as the one attached to a cuspidal automorphic representation of some $GL(n)$ over $\mathbb{Q}$) to a Galois orbit of algebraic integers in the complex plane (such as the Frobenius roots of abelian varieties over finite fields). I will explain a method for deriving no-shadowing root separation theorems in the latter context, yielding results that are sometimes sharp and better than the classical and general Mahler--Mignotte separation bound. We then discuss how to extend such a method to the former context, which is truly the one of interest in view of class number problems and the possibility of Landau--Siegel zeros. For the basic illustration, we explain how to prove that the abelian surfaces over $\mathbb{Q}$ whose L-function possesses an entire holomorphic square root are exactly the ones isogenous to the square of an elliptic curve.
16.30 - 17.00
Coffee and tea break
17.00 - 18.00 Anna Cadoret Centralizing sections and Q-compatibility (joint with Akio Tamagawa) Let $X$ be smooth variety over a number field and $Y\rightarrow X$ a smooth proper morphism. Consider the $l$-adic local system $V_l:=R^if_*\mathbb{Q}_l$, which we regard as a representation of the etale fundamental group of $X$. One expects (this follows e.g. from the Mumford-Tate conjecture or the unramified Fontaine-Mazur conjecture) that there is no closed point $x$ on $X$ such that the image of the corresponding decomposition group centralizes the image of the geometric etale fundamental group. I will explain how this problem can be reduced to prove the Q-compatibility of a certain family of $l$-adic sublocal systems $H_l$ of $V_l\otimes V_l^\vee$ and prove the Q-compatibly of the $H_l$ under a certain simplicity assumption.

Saturday, November 11

TimeSpeakerTitleAbstract
8.30 - 9.00
Coffee and tea break
9.00 - 10.00 Kaisa Matomäki Primes in short intervals and arithmetic progressions without L-functions I will discuss my on-going work with Jori Merikoski and Joni Teräväinen where we develop a new sieve argument which allows us to detect primes in sequences that have good type I information and satisfy certain additional conditions. Among other things, this allows us to give new L-function free proofs of Linnik's theorem on the least prime in arithmetic progression and of existence of primes in short intervals $[x, x+x^{1-\delta}]$ (with reasonable constants).
10.00 - 10.30
Coffee and tea break
10.30 - 11.30 Pierre Colmez A Kirillov model for completed cohomology I will explain how to construct a Kirillov model for Emerton's completed cohomology of the tower of modular curves and give some applications.