Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

## Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2017/18

Das Forschungsseminar findet dienstags in der Zeit von 13:15 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.

 24.10.2017 Enlin Yang (U Regensburg) Title: Characteristic class and the epsilon factor of an étale sheafAbstract: In this talk, we will briefly recall the definition and the properties of singular support and characteristic cycle of a constructible complex on a smooth variety. The theory of singular support is developed by Beilinson, which is motivated by the theory of holonomic D-modules. The characteristic cycle is constructed by Saito. In a joint work with Umezaki and Zhao, we prove a conjecture of Kato-Saito on a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field. In our proof, Beilinson and Saito's theory plays an essential role. 07.11.2017 Martin Lüdtke (U Frankfurt) Title: A birational anabelian reconstruction theorem for curves over algebraically closed fieldsAbstract: The question central to birational anabelian geometry is how strongly a field $K$ is determined by its absolute Galois group $G_K$. According to a conjecture of Bogomolov, if $K$ is the function field of a variety of dimension at least 2 over an algebraically closed field, it can by fully recovered from $G_K$. In dimension 1, however, $G_K$ is a profinite free group of rank equal to the cardinality of the base field, containing therefore no information about $K$. We show that a complete reconstruction is possible if one knows in addition how $G_K$ is embedded into the group of field automorphisms fixing only the base field. 28.11.2017 Ezra Waxman (U Tel Aviv) Title: Angles of Gaussian primesAbstract: Fermat showed that every prime p=1 mod 4 is a sum of two squares: p=a^2+b^2, and hence such a prime gives rise to an angle whose tangent is the ratio b/a. Hecke proved that these Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a conjecture, motivated by a random matrix model, for the variance of Gaussian primes across sectors, and discuss ongoing work about a more refined conjecture that picks up lower-order-terms. I will also introduce a function field model for this problem, which will yield an analogue to Hecke's equidistribution theorem. By applying a recent result of N. Katz concerning the equidistribution of "super even" characters (the function field analogues to Hecke characters), I will provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime. 05.12.2017 Efthymios Sofos (MPI Bonn) Title: An Erdős-Kac law for local solubility in families of varietiesAbstract: A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Daniel Loughran. 12.12.2017 Morten Risager (U Kopenhagen) Title: Arithmetic statistics of modular symbolsAbstract: Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols. We prove a refined version of this conjecture. 19.12.2017 Davide Cesare Veniani (Uni Mainz) Title: Recent advances about lines on quartic surfacesAbstract: The number of lines on a smooth complex surface in projective space depends very much on the degree of the surface. Planes and conics contain infinitely many lines and cubics always have exactly 27. As for degree 4, a general quartic surface has no lines, but Schur's quartic contains as many as 64. This is indeed the maximal number, but a correct proof of this fact was only given quite recently. Can a quartic surface carry exactly 63 lines? How many can there be on a quartic which is not smooth, or which is defined over a field of positive characteristic? In the last few years many of these questions have been answered, thanks to the contribution of several mathematicians. I will survey the main results and ideas, culminating in the list of the explicit equations of the ten smooth complex quartics with most lines. 16.01.2018 Walter Gubler (U Regensburg) Title: Non-archimedean Monge-Ampère equationsAbstract: We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of a metrized line bundle. We prove that the nonarchimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Ampère measure. Such a differentiability formula had been proposed by M. Kontsevich and Y. Tschinkel. In residue characteristic zero, it implies an orthogonality property for non-archimedean plurisubharmonic functions which allows us to drop an algebraicity assumption in a theorem of S. Boucksom, C. Favre and M. Jonsson about the solution to the non-archimedean Monge-Ampère equation. We will also present a similar result in positive equicharacteristic assuming resolution of singularities. 30.01.2018 Antareep Mandal (HU Berlin) Title: Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp formsAbstract: This talk updates our progress towards obtaining the optimal sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. In particular, we study the Maass operators and generalize the relation between the cusp forms and the Maass forms in higher dimensions along with presenting an interesting trick to obtain the heat kernel corresponding to the Maass Laplacians by adapting the "method of images" used to obtain the heat kernel corresponding to the Laplace-Beltrami operator. 06.02.2018 Xavier Roulleau (U Aix-Marseille) Title: Irrationality of cubic threefolds by their reduction mod 3 Abstract: A smooth cubic threefold X is unirational: there exists a dominant rational map f : P^3 → X. Clemens and Griffiths proved that a smooth complex cubic X is irrational, i.e. the degree of such f is always > 1. That was the first counter-example to the Lüroth Problem. The difficult part of their proof was to show that the intermediate Jacobian of X (which is an Abelian variety canonically attached to X) is not the Jacobian of a curve. In this talk we will prove that result anew for a generic cubic, by elementary methods: reduction mod p and point counting. This is a joint work with Dimitri Markouchevitch. 13.02.2018 Yuri Bilu (U Bordeaux) Title: Equations with singular moduli: effective aspectsAbstract: A singular modulus is the j-invariant of an elliptic curve with complex multiplication. André (1998) proved that a polynomial equation F(x,y)=0 can have only finitely many solutions in singular moduli (x,y), unless the polynomial F(x,y) is "special" in a certain precisely defined sense. Pila (2011) extended this to equations in many variables, proving the André-Oort conjecture on C^n. The arguments of André and Pila were non-effective (used Siegel-Brauer). I will report on a recent work by Allombert, Faye, Kühne, Luca, Masser, Pizarro, Riffaut, Zannier and myself about partial effectivization of these results.

Archiv:

Wintersemester 2007/08
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