Number Theory Seminar FS 2018

Unless otherwise stated, all talks start at 14.15 and take place in Seminarraum 05.002, Spiegelgasse 5

Information for speakers

01.03.2018Marta Pieropan (Ecole Polytechnique Fédérale de Lausanne)On rationally connected varieties over C_1 fields of characteristic 0In the 1950s Lang studied the properties of C_1 fields, that is, fields over which every homogeneous polynomial of degree at most n in n+1 variables has a nontrivial solution. Later he conjectured that every smooth proper rationally connected variety over a C_1 field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber-Harris-de Jong-Starr). This talk addresses the open case of Henselian fields of mixed characteristic with algebraically closed residue field.
15.03.2018Roger Baker (Brigham Young University)The second moment of the Dedekind zeta function on the critical lineFor the Riemann zeta function, and the Dedekind zeta function of a quadratic field, we know precisely how the second moment behaves on the critical line. This is not true when the degree of the algebraic number field is 3 or more. In the present talk I discuss a new upper bound for the second moment when the degree is at least 4.
22.03.2018Andrea Surroca On some conjectures on the Mordell-Weil and the Tate-Shafarevic groups of an Abelian varietyWe consider an Abelian variety defined over a number field. We give conditonal bounds for
1- the order of its Tate-Shafarevich group, as well as for
2- the Néron-Tate height of generators of its Mordell-Weil group.
The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals. We extend it to Abelian varieties of arbitrary dimension, defined over an arbitrary number field. In particular, with point 1- we improve and generalise a result by D. Goldfeld and L. Szpiro, and, with point 2- we extends a conjecture of S. Lang.
05.04.2018Will Sawin (Eidgenössische Technische Hochschule Zürich)The circle method and free rational curves on hypersurfacesIn joint work with Tim Browning, we study a pair of systems of Diophantine equations over F_q[t], and use the circle method to show a relationship between their numbers of solutions. As a consequence, we bound the dimension of the singular locus of the moduli space of rational curves on a smooth projective hypersurface. I will explain how these problems are related and what techniques we use to get the best bound.
19.04.2018Kevin Destagnol (Max Planck Institute for Mathematics)
17.05.2018Dan Loughran (The University of Manchester)
22.06.2018Francesco Amoroso (Université Caen Normandie)