Research Seminar Number Theory and Arithmetic Geometry

André--Oort for sums of powers in C^n

Pila's proof of André--Oort for C^n shows that the discriminants of the isolated special points on a hypersurface in C^n may be bounded by an ineffective constant that depends only on the degree of the hypersurface and the degree of its field of definition. In particular, the constant is independent of the height of the equation defining the hypersurface. Most of the special cases of André--Oort that are known effectively (due to Kühne, Bilu, Binyamini etc.) do not possess the same uniformity as Pila's ineffective result. In this talk, I will describe some results which are both uniform and effective for the family of hypersurfaces: a_1 x_1^m + ... + a_n x_n^m = b, where a_1, ..., a_n, b are rational and m is a positive integer.

Points of bounded height on del Pezzo surfaces

The Manin-Peyre conjecture predicts the distribution of points of bounded height on Fano varieties over number fields. I will report on joint work with Ulrich Derenthal where we study this conjecture in the case of del Pezzo surfaces of degree 5.

Fr 7.6.2024
14:00 Uhr

Diagonal quadric surfaces with a rational point

Following work of Serre on the solubility of conics, a current problem of interest in the study of Diophantine equations is to count the number of varieties in families which have a rational point.

In this talk I will give an overview of recent works in this area, before focussing on the particular example of diagonal quadric surfaces parameterised by {Y: wx=yz}. This family was first studied by Browning, Lyczak, and Sarapin, who showed that it exhibits an uncommonly large number of soluble members and attributed this phenomenon to the existence of thin sets on Y. They predicted that the “typical” behaviour should hold outside of these thin set, in the style of modern formulations of the Batyrev--Manin conjectures. In recent work I have shown that unusual behaviour occurs even with the removal of these thin sets by providing an asymptotic formula for the corresponding counting problem.

Finally, I will outline the character sum methods used to prove this result and introduce an adaptation of the large sieve for quadratic characters.

On quadratic twists of elliptic curves

The arithmetic of quadratic twist families of elliptic curves is an important topic in number theory, it is related to various Diophantine problems (e.g. congruent number problem, tiling number problem) and other aspects of number theory (e.g. recent modularity work by Caraiani and Newton). In this talk we propose an effectively computable parameter associated to a quadratic twist family of elliptic curves over Q with full rational 2-torsion points, which, if it is minimal, predicts that the density of the curves in this family with Mordell-Weil rank 0 is positive. In fact, this parameter determines the distribution of 2-Selmer groups of curves in this family completely.

Rational points of bounded height on a conic bundle surface over $\mathbb{F}_2(t)$

Loughran and Smeets stated a conjecture about the asymptotic behaviour of the number of varieties over number fields with rational points in certain families of proper, smooth algebraic varieties. We investigate whether their conjecture also holds over function fields.

We consider a conic bundle surface $C$ in $\mathbb{P}^2_K\times\mathbb{A}^1_K$  with a morphism $C\to\mathbb{A}^1_K$ over the global function field $K=\mathbb{F}_2((t))$. We are interested in an asymptotic formula for the number of fibres of $\mathbb{A}^1_K of bounded height that have a rational point. To obtain such an asymptotic formula, the main idea is to use harmonic analysis to compute the height zeta function and then use a Tauberian theorem for Dirichlet series with branch points.