Intercity Number Theory SeminarFebruary 7, Leiden. Location: Snellius building, morning room 401, afternoon room 412.
|11:15-12:15||Adelina Mânzăţeanu (UL), Rational curves on cubic hypersurfaces over 𝔽q|
Abstract. Using a version of the Hardy–Littlewood circle method over 𝔽q(t), one can count 𝔽q(t)-points of bounded degree on a smooth cubic hypersurface X ⊂ ℙn−1 over 𝔽q. Moreover, there is a correspondence between the number of 𝔽q(t)-points of bounded height and the number of 𝔽q-points on the moduli space which parametrises the rational maps of degree d on X. In this talk I will give an asymptotic formula for the number of 𝔽q-rational curves on X passing through two fixed points, one of which does not belong to the Hessian, for n ≥ 10, and q and d large enough. Further, I will explain how to deduce results regarding the geometry of the space of such curves.
|13:30-14:30||Adam Morgan (MPIM Bonn), Parity of Selmer ranks in quadratic twist families|
Abstract. We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we prove results about the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square.
|14:45-15:45||Rosa Winter (UL), Density of rational points on a family of del Pezzo surfaces of degree 1|
Abstract. Let k be a number field and X an algebraic variety over k. We want to study the set of k-rational points X(k). For example, is X(k) empty? And if not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are surfaces that are classified by their degree d (for d ≥ 3, these are the surfaces of degree d in ℙd). For all del Pezzo surfaces of degree ≥ 2 over k, we know that the set of k-rational points is dense provided that the surface has a k-rational point (that lies outside a specific subset of the surface for degree 2). But for del Pezzo surfaces of degree 1 over k, even though we know that they always contain a k-rational point, we do not know whether the set of k-rational points is dense. In this talk I will focus on a result that is joint work with Julie Desjardins, in which we prove that for a specific family of del Pezzo surfaces of degree 1 over k, under a mild condition, the k-rational points are dense with respect to the Zariski topology. I will compare this to previous results.
|16:00-17:00||Peter Koymans (MPIM Bonn), On Chowla’s conjecture over function fields|
Abstract. Let χ be a quadratic Dirichlet character. Then Chowla’s conjecture states that L(1/2, χ) is non-zero. Over function fields this conjecture has recently been disproved by Wanlin Li. The main result of this talk is that for many values of s we have that L(s, χ) is not zero for 100% of the characters χ. This is work in progress with Carlo Pagano and Mark Shusterman.
Intercity Number Theory SeminarMarch 20, Utrecht.
Intercity Number Theory SeminarApril 24, Groningen.
Belgian-Dutch Algebraic Geometry seminarMay 15, Leiden.
Intercity Number Theory SeminarJune 12, UvA and VU Amsterdam. At the UvA. Location: Science Park 904, room C0.110.
Intercity Number Theory Seminar / Getaltheorie in het vlakke landDecember 11, Utrecht.