Schedule for: 24w4002 - New Directions in Rational Points

I shall report on the Hasse principle for rational points on intersections of two quadrics in 7-dimensional projective space. The non-singular case was obtained by R. Heath-Brown in 2018 and revisited by me in 2022. The singular case was handled in 2023 by A. Molyakov.

In this talk we are interested in the average value of a multiplicative function when summed over a sequence that behaves well in small arithmetic progressions. As an application of our techniques, we obtain the size of the average 6-torsion, get tail bounds for the number of prime divisors of discriminants of $S_5$-extensions and count rational points on some varieties. This is joint work with Stephanie Chan, Carlo Pagano and Efthymios Sofos.

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov established the Hasse principle for Kummer varieties associated to 2- coverings of a principally polarised abelian variety A, under certain large image assumptions on the Galois action on A[2]. However, their method stops short of treating the case where the image is the full symplectic group, due to the possible failure of the Shafarevich--Tate group to have square order in this case. I will present recent work which overcomes this obstruction by combining second descent ideas in the spirit of Harpaz and Smith with new results on the 2- parity conjecture.

Let X/k be a variety over the field of fractions of a Henselian discrete valuation ring R. For example, k could be the field of p-adic numbers. I will explain how one can compute the set of all degrees of closed points on X from data pertaining only to the special fiber of a suitable model of X over R. In the case of curves over p-adic fields this gives an algorithm to compute the degree set, which yields some surprising possibilities. This is joint work with Bianca Viray.

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? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces. I will then explain how the reduction type (in particular, ordinary or non-ordinary good reduction) plays a role.

We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy-Littlewood circle method over number fields. This is joint work with Lillian Pierce and Damaris Schindler.

I discuss the number field analogue of a result by Green-Tao-Ziegler (2012) on linear patterns of prime numbers. This combined with techniques developed originally by Colliot-Thélène, Sansuc, Swinnerton-Dyer, Harari and others proves a Hasse principle type result for rational points on varieties over number fields fibered over P^1, as was done over Q by Harpaz-Skorobogatov-Wittenberg in 2014 using the Green-Tao-Ziegler theorem.

We show that the von Mangoldt functions for primes restricted to a fixed Chebotarev class or (conditionally on GRH) with a fixed primitive root are not correlated with nilsequences in a quantitative sense. Via Green-Tao-Ziegler nilpotent machinery, this yields asymptotic formulae for the number of solutions to non-degenerate systems of linear equations in such primes. This is joint work with Magdaléna Tinková.

We develop a two-dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q in at least 10 variables. This is a joint work with Pankaj Vishe (Durham) and Junxian Li (Bonn).

A conjecture of Malle predicts an asymptotic formula for the number of field extensions with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

We establish upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most 5 over global fields. The approach uses hyperplane sections and uniform upper bounds for the number of rational points of bounded height on elliptic curves. The results are unconditional in positive characteristic and for number fields rely on a conjecture relating the rank of an elliptic curve to its conductor. This is joint work with Leonhard Hochfilzer.

Let k\in \Z. I will compute the average size of the 2-Selmer group of y^2 = x^3 + B^k / \Q over B\in \Z.

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a smooth Fano variety, and it predicts an explicit asymptotic formula in terms of geometric invariants of the underlying variety. When you count rational points, it is important to exclude some contribution of rational points from an exceptional set so that the asymptotic formula reflects global geometry of the underlying variety. I will discuss applications of the study of exceptional sets to moduli spaces of sections of Fano fibrations, and in particular I will explain how exceptional sets explain pathological components of the moduli space of sections. This is based on joint work with Brian Lehmann and Eric Riedl.

Due to work of Manin and his collaborators we now have a good conjectural understanding of the distribution of rational points on Fano varieties. The analogous question of understanding the distribution of integral points on log Fano varieties has proven more challenging. In this talk I will discuss some of the new phenomena which appear when counting integral points and how one can understand them in terms of universal torsors. I will then explain how one can use universal torsors to count integral points on toric varieties. This corrects an unpublished preprint of Chambert-Loir and Tschinkel.

The existence of rational points on certain twisted moduli spaces of rank two stable vector bundles on curves over number fields has consequences for the existence of rational points on large dimensional quadrics over number fields. We explain a connection of this problem to a Hasse principle for the existence of large dimensional Grassmannian spaces in the intersection of two quadrics in the case of hyperelliptic curves. (Joint work with Jaya Iyer)

Classically, Diophantine approximation is the study of how well real numbers can be approximated by rational numbers with small denominators. However, there is an analogous question where we replace the real line with the real points of an algebraic variety: How well can we approximate a real point with rational points of small height? In this talk I will present joint work with Zhizhong Huang and Damaris Schindler in which we study this question for projective quadrics. Our approach makes use of a version of the circle method developed by Heath-Brown, Duke, Friedlander and Iwaniec.

Stacks are ubiquitous in algebraic geometry and in recent years there has been increased interest in studying the arithmetic of stacks and using their structure to answer more classical questions in number theory. The classical problem of counting elliptic curves with a rational N-isogeny can be phrased in terms of counting rational points on certain moduli stacks of elliptic curves. I will talk about the recent progress that has been made on this problem within this context, as well as some open questions in connection with the stacky Batyrev-Manin-Malle conjecture. The talk assumes no prior knowledge of stacks and is based on joint work with Brandon Boggess.

Let E be a field and G be an adjoint classical group defined over E. Let G(E) denote the group of E-rational points of G and let G(E)/R denote the R-equivalence classes. We discuss the triviality of G(E)/R over fields with low virtual cohomological dimension.

Let k be a field and let V be a linear and faithful representation of a finite group G. The Noether problem asks whether V/G is a (stably) rational variety over k. It is known that if p=char(k)>0 and G is a p-group, then V/G is always rational. On the other hand, Saltman and later Bogomolov constructed many examples of p-groups such that V/G is not stably rational over the complex numbers. The aim of the talk is to study what happens over a discrete valuation ring R of mixed characteristic (0,p). We show for instance that for all the examples found by Saltman and Bogomolov, there cannot exist a smooth projective scheme over R whose special, respectively generic fibre are stably birational to V/G. The proof combines integral p-adic Hodge theory and the study of differential forms in positive characteristic.

In joint work with Damaris Schindler we develop a new version of the hyperbola method for counting rational points of bounded height that generalizes the work of Blomer and Brüdern for products of projective spaces. The hyperbola method transforms a counting problem into an optimization problem on certain polytopes. For rational points on subvarieties of toric varieties, the polytopes have a geometric meaning that reflects Manin's conjecture, and the same holds for counts of Campana points of bounded height. I will present our results as well as some general heuristics.

Given an elliptic curve E over Q of conductor N, there exists a surjective morphism from X_0(N) to E defined over Q. The modular degree m_E of E is the minimum degree of all such modular parametrizations. Watkins conjectured that the rank of E(Q) is less than or equal to \nu_2(m_E). In this talk, we shall delve into various analytic approaches, aiming to deepen our understanding and address this intriguing conjecture.

Let K be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of H^1(K,G) \to \prod_{v \in M_K} H^1(K_v,G). For a general G, there is a Brauer-Manin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the only one) for supersolvable groups. I will present a new fibration theorem over quasi-trivial tori that, combined with the approach of Harpaz and Wittenberg, gives a positive answer for all solvable groups. The fibration theorem presents two difficulties: lifting local points and avoiding Brauer-Manin obstruction on the fibers. To overcome the first, we employ ideas of Shafarevich used in his solution of the Inverse Galois Problem for solvable groups. To overcome the second, one first reduces to a desirable subcase using a base-change method due to Harpaz and Wittenberg. One then proceeds with the core computation of the "triple variation'' of the Brauer-Manin obstruction on the fibers in terms of some Redéi symbols (which may be thought as “triple pairings” of algebraic numbers) and concludes by following a general combinatorial principle first noted by Alexander Smith in the context of class groups and Selmer groups. This is work in progress. Partial results (including the "lifting local points” part and the appearance of “triple pairings” in the BMO) were also obtained independently by Harpaz and Wittenberg.