### Number Theory Seminar FS 2017

Unless otherwise stated, all talks start at 14.15 and take place in seminar room 00.003, Spiegelgasse 1

Information for speakers

Date | Speaker | Title | Abstract |
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02.03.2017 | Multiplicatively dependent algebraic numbers | (joint work with Alina Ostafe, Francesco Pappalardi, Min Sha, Cam Stewart) We discuss various questions related to the distribution of vectors of algebraic numbers (u_1, …., u_n) which are multiplicatively dependent. In particular, we present some counting results for the number of such vectors of degree d and height h (from a fixed number field K and from Q-bar). We also give both sided estimates on their density in R^n. Finally we give an analogue of a result of Bombieri-Masser-Zannier (1999) proving the boundedness of the “house" of the shifts v from the abelian closure of a given number field K, for which the vector (u_1-v, …., u_n-v) becomes multiplicatively dependent, rather than the boundedness of their height as in BMZ’99 (but for shifts v from Q-bar). This result has applications to multiplicative dependence in orbits of polynomial dynamical systems, generalising those on roots of unity. | |

06.04.2017 | Erratic behavior of the coefficients of modular forms | I will speak on a recent joint work with Jean-Marc Deshouillers, Sanoli Gun and Florian Luca. Here is a sample result. Let $\tau(.)$ be the classical Ramanujan $\tau$-function defined by $$q\prod_{n>0} (1-q^n)^{24} = \sum_{n>0} \tau(n) q^n.$$ The classical work of Rankin implies that both inequalities $|\tau(n)|<|\tau(n+1)|$ and $|\tau(n)|>|\tau(n+1)|$ hold for infinitely many $n$. We generalize this for longer segments of consecutive values of $\tau$. Let k be a positive integer such that $\tau(n)$ is not $0$ for $n\le k/2$. (This is known to be true for all $k < 10^{23}$, and, conjecturally, holds for all $k$.) Let s be a permutation of the set $\{1,...,k\}$. Then there exist infinitely many positive integers $n$ such that $|\tau(n+s(1))|<\tau(n+s(2))|<...<|\tau(n+s(k))|$. | |

11.04.2017 | Subgroups of Class Groups and the Absolute Chevalley-Weil Theorem | The following conjecture is widely believed to be true: given a finite abelian group G, a number field K and an integer d>1, there exist infinitely many extensions L/K of degree d such that the class group of L contains G as a subgroup. I will speak on some old and recent results on this conjecture, in particular, on my recent joint work with J. Gillibert.
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20.04.2017 | Lehmer's problem on Drinfeld modules | This talk will be about lower bounds for the canonical height associated to a Drinfeld module. I will explain how one can obtain such a lower bound for the canonical height of an algebraic point which is polynomial in the inverse of the degree of the point. This bound is valid for every Drinfeld module (in particular, of arbitrary rank). This is a joint work with Aurélien Galateau. | |

11.05.2017 | Unlikely intersections in families of abelian varieties and some polynomial Diophantine equations | I will report on some results about unlikely intersections in families of abelian varieties, obtained mostly in collaboration with L. Capuano, and explain how they fit into the framework of the Zilber-Pink Conjectures. Finally, I will present some applications concerning certain polynomial Diophantine equations. | |

18.05.2017 | On the action of the absolute Galois group on curves and surfaces | A combination of Hodge theory and Serre's GAGA principle implies that many topological invariants of complex projective varieties, such as Betti numbers and Chern classes, remain invariant under Galois action. Nevertheless, in 1964 Serre proved that conjugate varieties need not be homeomorphic, by showing the existence of Galois elements $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ and projective varieties $X$ such that the fndamental groups of $X$ and its conjugate variety $X^{\sigma}$ are not isomorphic (while their algebraic fundamental groups are). In this talk I will attempt to show that this occurs for every $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ different from the identity and the complex conjugation. Our projective varieties $X$ will be Beauville surfaces, a kind of rigid surfaces of general type introduced by Catanese, arising as finite quotients of products of triangle curves. This is joint work with Andrei Jaikin-Zapirain. | |

01.06.2017 | Transcendental Liouville inequality on projective variety and rational points on analytic disks | Let $(X, L)$ be a polarized projective variety defined over a number field. If $D$ is a one dimensional analytic disk in $X(\mathbf{C})$ then, a classical theorem of Bombieri and Pila tells us that there are at most $\exp(\epsilon T)$ rational points of height at most $T$ in $D\cap X(K)$. Following an idea due to Masser, one can see that, if $D$ contains some special transcendental point, then the cardinality of the set of points of height at most $T$ in $D\cap X(K)$ is polynomial in $T$. These special points, called points of type $S$, verify a property which is similar to Liouville inequality. We will explain why points of type $S$ are full $X(\mathbf{C})$ (for the Lebesgue measure). |