New directions in rational points
Titles and
abstracts of talks
Levent Alpöge
The average size of 2-Selmer
groups in the families y^2 = x^3 + B^k
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.
Subham Bhakta
Arithmetic
statistics of modular degree
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.
Tim Browning
Generalised quadratic forms over totally real
number fields
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.
Jean-Louis Colliot-Thélène
Arithmetic on the intersection of
two quadrics
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.
Brendan Creutz
Degrees of points on
varieties over Henselian fields
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.
Julian Demeio
The Grunwald Problem
for solvable groups
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.
Christopher Frei
Linear
equations in Chebotarev and Artin primes
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.
Jakob Glas
Rational
points on del Pezzo surfaces of low degree
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.
Wataru Kai
Linear patterns of prime
elements in number fields
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.
Peter Koymans
Averages
of multiplicative functions over integer sequences
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.
Dan Loughran
The leading
constant in Malle's conjecture
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.
Adam
Morgan
On the Hasse
principle for Kummer varieties
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.
Simon Rydin Myerson
A two-dimensional delta method and
applications to quadratic forms
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).
Margherita Pagano
The role of primes of good reduction in the
Brauer-Manin obstruction to weak approximation
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.
Raman Parimala
A Hasse principle for twisted
moduli spaces
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)
Marta Pieropan
Points of bounded height on certain subvarieties of
toric varieties
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.
Raman Preeti
R-equivalence in adjoint classical groups
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.
Soumya Sankar
Counting points on stacks and elliptic
curves with a rational N-isogeny
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.
Tim
Santens
Manin’s conjecture for integral points on toric varieties
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.
Alec Shute
Zooming in on quadrics
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.
Sho Tanimoto
Sections of Fano fibrations over curves
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.
Domenico
Valloni
Noether’s problem in mixed
characteristic
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.