### Number Theory Seminar FS 2016

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

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

25.02.2016 | Reductions of abelian varieties | Let A and A’ be two abelian varieties defined over a number field. Our goal is comparing A and A', for example we would like to know if they are isogenous. We consider all reductions, and the information at our disposal are the number of points over the residue fields or the size of the reductions of the Mordell-Weil group. This is joint work with Chris Hall (University of Wyoming). | |

03.03.2016 | Around the Property (B) | The Property (B) was introduced 15 years ago by Bombieri and Zannier to study fields of algebraic numbers on which the Weil height can be bounded from below (outside of the roots of unity). This property is linked to a relative version of the Lehmer problem, which plays an important role in diophantine geometry, for instance towards the Zilber-Pink conjecture. We'll discuss examples that have been found recently of fields with the Property (B). | |

10.03.2016 | Effective Pila--Wilkie bounds for restricted Pfaffian surfaces | The counting theorem of Pila and Wilkie has become celebrated for opening up one of the most important developments in applied model theory in recent years. It provides a bound on the density of rational points of bounded height lying on the `transcendental parts' of sets definable in o-minimal expansions of the real field, a result which has had several stunning number theoretic applications (e.g. to the Manin-Mumford and André-Oort Conjectures). However, the proof of the theorem is not effective: it does not give a procedure which, given a definable set, will compute the Pila--Wilkie bound for that set. This of course constrains the effectivity of its applications. I will discuss some recent progress made towards finding an effective version of the Pila--Wilkie Theorem in certain cases. (Joint work with G. O. Jones.) | |

17.03.2016 | Around the Northcott property | Motivated by the question of counting periodic points of endomorphisms of an abelian variety, Northcott proved in 1949 that every number field contains just finitely many points of bounded height. There are also fields of infinite degree over $\mathbb{Q}$ satisfying this finiteness property. We say that such fields satisfy the Northcott property (N). In this talk we will discuss relations between (N) and other arithmetic properties a field $F\subseteq\overline{\mathbb{Q}}$ may have. | |

31.03.2016 | Spaces generated by products of two Eisenstein series. | Which modular forms are linear combinations of products of two Eisenstein series? Writing a modular form as such a linear combination can be used in many applications, e.g. for calculating Fourier expansions at any cusp. In joint work with Martin Dickson we show that for many congruence subgroups all modular forms are linear combinations of products of Eisenstein series. I will present a proof of this result using the Rankin-Selberg method and modular symbols. In the end I will talk about recent work by Rogers-Zudilin and Brunault on some of Boyd's conjectures concerning relations between Mahler measures and L-values of elliptic curves. They write a newform of weight 2 as a linear combination of products of Eisenstein series in order to study the L-value L(f,2). I will indicate how this approach can be generalised to study the L-value of a K3 surface and connect it to Mahler measures. | |

08.04.2016 | On the Hilbert Property and the fundamental group of algebraic varieties | This would concern recent work with P. Corvaja in which we relate the Hilbert Property (a kind of axiom linked with Hilbert Irreducibility) for an algebraic variety with its fundamental group. This leads to new examples (of surfaces) both of validity and failure of the Hilbert Property. | |

26.04.2016 | Height of rational points on algebraic varieties | Given a finite set S of irreducible homogeneous polynomials in several variables and with integral coefficients, a classical problem in number theory is to describe the set V(Q) of rational points on the variety V defined by S, that is the set of common zeros with rational coordinates of the polynomials in S. Typical questions are: when is V(Q) non-empty or infinite? If V(Q) is infinite, can we measure its size? If V(Q) is non-empty, what is the size of its smallest element? We will be concerned in particular with the latter problem. While all these questions are too hard in such generality, we will try to explain how they become easier when investigated on average over families of varieties. | |

19.05.2016 | The Distribution of Slopes of Gaussian Primes for the Rational Function Field | Hecke proved that the 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 function field analogue to this result, by making use of "super even" characters and their associated L-functions. By applying a recent result of N. Katz concerning the equidistribution of super even characters, I will also 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. | |

26.05.2016 | Compatible systems of $\ell$-adic representations arising from abelian varieties. | Famous (and still unproven in full generality) conjectures of Serre-Grothendieck, Tate and Fontaine-Mazur describe $\ell$-adic representations that arise from the action of the absolute Galois group of a number field $K$ on the (twisted) $\ell$-adic cohomology groups of projective algebraic varieties that are defined over $K$. Assuming all these conjectures (and the Hodge conjecture), we discuss the following question: which $\ell$-adic representations correspond to the $\ell$-adic Tate modules of an abelian variety? We give an answer for abelian varieties without nontrivial endomorphisms. This is a report on a joint work with Stefan Patrikis and Felipe Voloch. | |

02.06.2016 | Two reformulations of the Lehmer conjecture | Salem numbers are those numbers (real >1) with at most one conjugate outside the unit disc and at least one on the unit circle. Salem's conjecture asserts that they cannot be too close to 1. More generally, Lehmer's conjecture asserts that given any algebraic unit, which is not a root of unity, the modulus of the product of the conjugates lying outside the unit disc (the Mahler measure) is bounded away from 1. I will present two joint works, one with B. Deroin and the other with P. Varju, in which we give two unrelated ways to reformulate Salem's and Lehmer's conjecture. The first as a uniform spectral gap for a certain family of hyperbolic surfaces, the second as an elementary counting problem in finite fields. | |

09.06.2016 | Polarized isogeny orbits in families of abelian varieties | Abstract: Let $\mathcal{A}$ be an abelian scheme over $B$. Fix a point $s$ in $\mathcal{A}$ and let $\Sigma$ be the polarized isogeny orbit generated by $s$. We characterize curves in $\mathcal{A}$ whose intersections with $\Sigma$ is Zariski dense and give the conjecture for higher dimensional subvarieties. |