### Number Theory Seminar HS 2015

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

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

17.09.15 | The Lang-Vojta conjecture and arithmetic finiteness results for smooth hypersurfaces | In 1983, Faltings proved the Shafarevich conjecture: for a finite set of finite places of a number field K and an integer g>1, the set of isomorphism classes of curves of genus g over K with good reduction outside S is finite. In this talk we shall consider analogues of the Shafarevich conjecture for hypersurfaces. We will prove, assuming the conjecture of Lang-Vojta, the analogous finiteness statement for hypersurfaces of fixed degree and fixed dimension. Unconditionally, we prove the Shafarevich conjecture for hypersurfaces of Hodge level at most one, and some hypersurfaces of Hodge level 2. This is joint work with Daniel Loughran. | |

09.10.15 | Galois groups of logarithmic equations | I will describe a recent joint work with A. Pillay, where we extend to semi-abelian schemes the classical theorem of Ax on the exponential of algebraic functions. The proof is based on my co-author's theory of logarithmic differential equations, combined with an argument of Galois descent reminiscent of Kummer theory | |

15.10.15 | Multiplicative and modular diophantine problems | I will describe some diophantine results and conjectures, from the Mordell conjecture of 1922 (theorem of Faltings) to the open and very general Zilber-Pink conjecture. I will describe a recent result and conjecture of similar flavour which are not formally consequences of the Zilber-Pink conjecture. | |

22.10.15 | A solution of a problem of Ramanujan of 1915 | It is well known that the number of divisors $d(n)$ is large, when $n$ is a product of many primes. Wigert determined the maximal order of magnitude of the divisor function: \[d(n) \leq \exp ( (\log 2 +o(1)) \frac{\log n}{\log \log n},\] or in other words \[ \max_{n\le x} \log d(n) \sim (\log 2){\frac{\log x}{\log \log x}}. \] Ramanujan (1915) was the first to investigate the maximal order of magnitude of the iterated divisor function $d(d(n))$, giving an example that \[ d(d(n) \ge (\sqrt{2}\log 4 - o(1)) \frac{\sqrt{\log n}}{\log \log n}\] for infinitely many $n$. Erd\H{o}s, K\'{a}tai, Ivi\'{c} and Smati gave upper bounds, but determining an analogue of Wigert's result, i.e. a best possible upper bound, was an open problem. In this talk we give a solution to this problem: \[ \max_{n\le x} \log d(d(n))= \frac{\sqrt{\log x}}{\log_2 x} \left( c + O(\frac{\log_3 x}{\log_2 x} )\right), \] where \[ c =\Bigg( 8 \sum_{j=1}^\infty \log^2 (1+1/j) \Bigg)^{1/2} = 2.7959802335\ldots. \] (Joint work with Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, Jan-Christoph Schlage-Puchta.) | |

29.10.15 | An explicit Andr\'e-Oort type result for P^1(C) x G_m(C). | We will discuss a problem of Andr\'e-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$ are of the form $(\alpha, \lambda)$, with $\alpha$ a singular modulus and $\lambda$ a root of unity. The qualitative version of our result states that if $\mathcal{C}$ is a closed algebraic curve in $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$, defined over a number field, not containing a horizontal or vertical line, then $\mathcal{C}$ contains only finitely many special points. We discuss two approaches, one using logarithmic forms, and another using class field theory. Both approaches give explicit results. | |

12.11.15 | p-adic measures of Hermitian modular forms and the Rankin-Selberg method | p-adic measures are playing an important role in the various Main Conjectures of Iwasawa Theory. In this talk I will start by presenting some basic properties of the classical L functions associated to a Dirichlet character, such as the Kummer congruences, and then explain how these properties can be understood in a broader context, namely that of the existence of a p-adic measure. Then after discussing some basics of Hermitian modular forms, (automorphic forms associated to unitary groups) I will present the construction of various p-adic measures, which can be associated to a Hermitian modular form by employing the so-called Rankin-Selberg method. | |

03.12.15 | Gamma values: regular and irregular | The values of the gamma function at rational numbers remain quite mysterious, one of the reasons being that (conjecturally) they are not periods in the usual sense of algebraic geometry. However, the theory of regular singular connections allows to show that suitable products of them are periods of Hodge structures with complex multiplication, as predicted by Gross and Deligne. To deal with single gamma values, one needs to consider irregular singular connections instead. After an exposition of my results on the Gross-Deligne conjecture, I will explain how irregular singular connections may shed some light on the arithmetic nature of these numbers. | |

10.12.15 | Algebraic structures of exponential maps | I will discuss the problem of giving a complete algebraic description of the interaction between algebraic geometry and the exponential map of a complex algebraic group, focusing on the case of a simple abelian variety. The problem and many of the tools originate in model theory, but I will concentrate more on the role played by transcendence theory and the Faltings-Ribet Kummer theory for abelian varieties. The talk will be based on recent work with Jonathan Kirby, extending work of Zilber on the case of the multiplicative group. |