### 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

Date | Speaker | Title | Abstract |
---|---|---|---|

01.03.2018 | On rationally connected varieties over C_1 fields of characteristic 0 | In 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. | |

15.03.2018 | The second moment of the Dedekind zeta function on the critical line | For 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. | |

22.03.2018 | On some conjectures on the Mordell-Weil and the Tate-Shafarevic groups of an Abelian variety | We 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. | |

05.04.2018 | The circle method and free rational curves on hypersurfaces | In 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. | |

19.04.2018 | |||

17.05.2018 | |||

22.06.2018 |