Intercity Number Theory Seminar


Intercity seminar

22 October, Leiden. Sylvius laboratory room 1.4.31, starting at 11:15
Remy van Dobben de Bruyn Leiden, Equivalent conjectures on independence of ℓ
The ℓ-adic étale cohomology of a variety is defined in terms of the geometry of ℓ-power degree coverings. Unlike in algebraic topology, there is no integer-valued theory to compare with, leading to a plethora of “independence of ℓ” conjectures for varieties over finite fields. We will survey what is known, and prove that independence of ℓ for Betti numbers is equivalent to two other statements involving algebraic cycles.
Margherita Pagano Leiden, An example of a Brauer–Manin obstruction to weak approximation at a prime with good reduction
A way to study rational points on a variety is by looking at their image in the p-adic points. Some natural questions that arise are the following: is there any obstruction to weak approximation on the variety? Which primes might be involved in it? Bright and Newton have proven that for K3 surfaces defined over number fields primes with good ordinary reduction play a role in the Brauer–Manin obstruction to weak approximation.

In this talk I will give an explicit example of this phenomenon. In particular, I will exhibit a K3 surface defined over the rational numbers having good reduction at 2, and for which 2 is a prime at which weak approximation is obstructed.

Daan van Gent Leiden, Indecomposable algebraic integers
In algebraic number theory, the finiteness of the Picard group of an order in a number field is generally proved via a lattice argument: the order forms a lattice and every ideal class contains an integral ideal with a small enough non-zero element. In this talk we will consider ℤ̄, the ring of algebraic integers, which is a lattice in a similar sense, and we will treat this lattice as intrinsically interesting. We will state several open problems and some partial results, mainly with regards to the Voronoi polyhedron of ℤ̄. This motivates the study of the indecomposable elements of ℤ̄.
Pim Spelier Leiden, Geometric quadratic Chabauty on X0(67)+
By Faltings’ theorem, a smooth, projective curve of genus greater than 1 has finitely many rational points. A common method to try and find all these rational points is Chabauty’s method, which embeds the curve in another variety, traditionally the Jacobian of the curve, where the rational points are known. Recently, Edixhoven and Lido extended this method to geometric quadratic Chabauty, where the extra data of an endomorphism of the Jacobian is used to construct a more interesting variety for Chabauty’s method.

Intercity seminar

12 November, Leiden. Snellius building room 401, starting with coffee/tea at 11:00. Live stream via Zoom
Peter Stevenhagen Leiden, Elliptic curves and primes of cyclic reduction
Let E be an elliptic curve defined over a number field K. Then for every prime p of K for which E has good reduction, the point group of E modulo p is a finite abelian group on at most 2 generators. If it is cyclic, we call p a prime of cyclic reduction for E. We will answer basic questions for the set of primes of cyclic reduction of E: is this set infinite, does it have a density, and can such a density be computed explicitly from the Galois representation associated to E?

This is joint work with Francesco Campagna (MPIM Bonn).

Charlotte Dombrowsky Leiden, An upper bound for packings on spheres
In 1611 Kepler made his famous conjecture about packings of three-dimensional spheres in Euclidean space: he described two arrangements of spheres and suggested that no other arrangement has a greater average density. As difficult as it was to prove this conjecture – the formal proof was only accepted in 2017 – as easy it is to generalize this problem to higher dimensions: The n-dimensional sphere packing problem is the question about the densest regular arrangement of n-dimensional spheres in n-dimensional Euclidean space.

In this talk, we will study at a slightly different packing problem: Instead of looking at packings of spheres, we look at packings on spheres. Given an object on the n-dimensional sphere, what is the maximum number of non-intersecting copies of this object that we can place on the sphere? The core of our approach is to use a connection between graphs and packings: a positive definite function with certain properties defined on the group of symmetries of the sphere provides an upper bound for the independence set of the packing graph and thus for the packing. To generate such a function, we study unitary, irreducible representations and in particular the quasi-regular representation of SO(n) restricted to the space of harmonic homogeneous functions of fixed degree.

Yukako Kezuka MPIM Bonn, On central L-values of twists of the Fermat elliptic curve
I will study elliptic curves CN over ℚ of the form x3+y3=N for any cube-free positive integers N, allowing N to be divisible by 2 or 3. They are cubic twists of the Fermat elliptic curve x3+y3=1, and they admit complex multiplication by the ring of integers of the imaginary quadratic field ℚ(√−3). I will obtain a lower bound for the 3-adic valuation of the algebraic part of their central L-values. I will then show that the bound is sharp in certain special cases, which gives us the 3-part of the conjecture of Birch and Swinnerton-Dyer for CN/ℚ in these cases.
Eugenia Rosu TU Darmstadt / Leiden, Twists of elliptic curves with complex multiplication
We consider certain families of sextic twists of the elliptic curve y2=x3+1 that are not defined over ℚ, but over ℚ(√−3). We compute a formula that relates the value of the L-function L(ED, 1) to the square of a trace of a modular function at a CM point. Assuming the Birch and Swinnerton-Dyer conjecture, when the value above is non-zero, we should recover the order of the Tate-Shafarevich group, and under certain conditions, we show that the value is indeed a square.

Intercity seminar / Getaltheorie in het Vlakke Land / Arithmétique en Plat Pays

10 December, Utrecht. Cancelled
Laurent Berger ENS Lyon, Unlikely intersections for formal groups
Michel Rigo Liège, Numeration systems: a link between formal languages and number theory
Amita Malik MPIM, Integer partitions analysis
Lola Thompson Utrecht, Salem numbers and short geodesics