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 (GraberHarrisde JongStarr). 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 MordellWeil and the TateShafarevic 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 TateShafarevich group, as well as for 2 the NéronTate height of generators of its MordellWeil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and SwinnertonDyer conjecture and the functional equation of the Lseries. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the nontorsion 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 extend 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.  
12.04.2018  Unlikely intersections in families of semiabelian surfaces  For an elliptic curve $E$ the Poincar\`e biextension of $E$ parametrizes points on $\mathbb{G}_m$ extensions of $E$. It can be thought of as a group scheme $G$ over the dual $\hat{E}$ of $E$. I will talk about joint work with Daniel Bertrand respectively Gareth Jones on qualitative results concerning the ZilberPink conjecture for $G$ respectively uniform and effective results for the relative ManinMumford question.  
19.04.2018  Prime and squarefree values of polynomials in moderately many variables  The classical Schinzel's hypothesis and its quantitative version, the BatemanHorn's conjecture, states that a system of polynomials in one variable takes infinitely many simultaneously prime values under some necessary assumptions. We will present in this talk a proof of a generalization of these conjectures to the case of a integer form in many variables. In particular, we will establish that a polynomial in moderately many variables takes infinitely many prime (but also squarefree) values under some necessary assumptions. The proof relies on the Birch's circle method but can be achieved in 50% fewer variables than in the classical Birch setting. Moreover this result can be applied to study the Hasse principle and weak approximation for some normic equations.  
17.05.2018  Probabilistic arithmetic geometry  A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos.  
31.05.2018  An analogue of the BrauerSiegel theorem for an ArtinSchreier family of elliptic curves  The classical BrauerSiegel theorem gives upper and lower bounds on the product of the classnumber times the regulator of units of a number field, in terms of its discriminant. Consider now an elliptic curve E defined over F_q(t): one can form the product of the order of the TateShafarevich group of E (assuming it is finite) and of its NéronTate regulator. In analogy with the above theorem, we are interested in finding good upper and lower bounds of this quantity in terms of simpler invariants of E, e.g. its height. In general such bounds are hard to obtain (in particular, the lower bounds are the most delicate); a satisfactory answer to the question is known only for a handful of cases. In this talk, I will report on a recent work where I studied an "ArtinSchreier family" of elliptic curves. I will explain how good unconditional bounds can be found in this case. This provides a new example of family of elliptic curves for which an analogue of the classical BrauerSiegel theorem holds.  
07.06.2018  NewtonOkounkov bodies of exceptional curve valuations  Let $p$ be a closed point in ${\mathbb{P}}_{\mathbb{C}}^2$ and consider a surface obtained by a sequence of finitely many blowups of points where we start with $p$ and always blow up a point in the exceptional divisor created last. Our result is an explicit description of the NewtonOkounkov body of the pullback of $\mathcal{O}_{\mathbb{P}^2}(1)$ with respect to the flag given by the last exceptional divisor and a point on it. This is joint work with Carlos Galindo, Francisco Monserrat and Julio José MoyanoFernández.
 
20.06.2018  A variation of heights and small points in families of elliptic curves  Bogomolov's conjecture (now a theorem following work of Zhang and Ullmo) concerns the geometry of points with small canonical height on abelian varieties. In joint work with Laura DeMarco, we prove an analogous result in the setting of families of products of elliptic curves. This extends theorems by Masser and Zannier in the theme of unlikely intersections. A key ingredient in our proof is work of Silverman concerning the variation of heights in families of elliptic curves. In this talk, I will discuss this result and its applications towards a Bogomolovtype extension of a recent result by Barroero and Capuano, which extends the aforementioned result of Masser and Zannier. The latter is work in progress joint with Laura DeMarco.
 
22.06.2018  Height on Galois extension  In a recent joint paper with D. Masser we prove that the Weil height of a nonzero algebraic number, not a root of unity, which generates a Galois extension, can be bounded from below "essentially" by a positive constant. We further analyse Galois extension with full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of "big" height. These results answer a question of C. Smyth and provide some evidence to a conjecture which asserts that the height of such a generator growth to infinity with the degree of the extension.
