BAGS: Jan - Jun 2024. Organisers:  Julian Demeio, Harry Shaw, and Harkaran Uppal.

Speaker: Julie Tavernier (University of Bath) - 18th June 2024

Title: Counting number fields whose conductor is the sum of two squares

Abstract: In this talk we will look at the number of abelian field extensions of a number field with fixed Galois group G whose conductor is the sum of two squares. We will employ techniques from harmonic analysis and class field theory, and use a Poisson summation formula due to Frei, Loughran and Newton to find an asymptotic for the number of these number fields. We will then introduce the theory of frobenian multiplicative functions and state a more general result about counting number fields with frobenian conditions imposed on the conductor.

Speaker: Peter Koymans (ETH Zurich) -  4th June 2024

Title: Sums of rational cubes and the 3-Selmer group

Abstract: Alpoge--Bhargava--Shnidman recently found the average size of the 2-Selmer group in the cubic twist family x^3 + y^3 = n. Using completely different techniques, we compute the distribution of the 3-Selmer group, improving their bounds for the proportion of integers that are/are not sums of rational cubes. The main new input is a trilinear sieve for "generalized Redei symbols". The goal of this talk is to explain these new generalized Redei symbols, their importance and some ideas behind the new sieve. This is joint work with Alexander Smith.

Speaker: Christopher Keyes (King's College London) -  21st May 2024 

Title: How often does a cubic hypersurface have a point?

Abstract: A cubic hypersurface in projective n-space defined over the rationals is given by the vanishing locus of an integral cubic form in n+1 variables. For n at least 4, it is conjectured that these varieties satisfy the Hasse principle. Recent work of Browning, Le Boudec, and Sawin shows that this conjecture holds on average, in the sense that the density of soluble cubic forms is equal to that of the everywhere locally soluble ones. But what do these densities actually look like? We give exact formulae in terms of the probability that a cubic hypersurface has p-adic points for each prime p. These local densities are explicit rational functions uniform in p, recovering a result of Bhargava, Cremona, and Fisher in the n=2 case, as well as the fact that all cubic forms are everywhere locally soluble when n is at least 9. Consequently, we compute numerical values (to high precision) for natural density of cubic forms with a rational point for n at least 4, and a conjectural value for n=3, the case of cubic surfaces. This is joint work with Lea Beneish.

Speaker: Efthymios Sofos (University of Glasgow) - 23rd April 2024

Title: Diophantine stability for curves of genus 0

Abstract: In joint work with Carlo Pagano we show that genus 0 curves are Diophantine stable with probability 1. The error term exhibits nothing like a square-root cancellation but instead involves quantities usually seen in the context of Manin's conjecture. The proof involves a common generalisation of the geometric and the large sieve.

Speaker: Ross Paterson (University of Bristol) - 26th March 2024

Title: Quadratic Twists as Random Variables

Abstract: Let E/Q be an elliptic curve and K be a quadratic field of discriminant d. It is well known that the rank of E(Q) and the rank of the quadratic twist E_d(Q) sum to the rank of E(K) — but do these groups generate E(K)? The answer to this question is sometimes yes and sometimes no, which leads to the natural question: how common is each answer? We will provide partial answers, in particular showing that one answer is a lot more likely than the other!

In studying this question we are drawn to consider the behaviour of the 2-Selmer groups of E and E_d. There is a heuristic model of Poonen-Rains for the behaviour of these 2-Selmer groups individually, as E varies, but how independent are they? We'll describe a heuristic in this direction, and some results in support of it.

Speaker: Damián Gvirtz-Chen (University of Glasgow) - 12th March 2024

Title: How many rational points does a diagonal surface have?

Abstract: Using a topological deformation argument, we determine the Brauer group of surfaces defined by an equality f=a*g of two monic polynomials f and g, provided the leading coefficient a is generic. E.g. the generic diagonal surface of degree d (with indeterminate coefficients) has trivial Brauer group. However, the general diagonal surface of degree d over Q(zeta_d) has cyclic algebraic Brauer group of order d or d/2 depending on the parity of d. It follows that weak approximation fails for 100% of everywhere locally soluble diagonal surfaces.

(joint work with A.N. Skorobogatov)

Timothy Browning (ISTA) - 27th February 2024

Title: 100% of quadratic twists have no integral points

Abstract: In recent joint work with Stephanie Chan, for a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses the reduction theory of binary quartic forms, bounds for rational points of bounded height on certain singular cubic surfaces, and Heath-Brown’s character sum analysis of Selmer groups for the congruent number curve.

Martin Orr (University of Manchester) - 13th February 2024

Title: Isogeny coincidences between families of elliptic curves

Abstract: There are only finitely many algebraic numbers t such that the three elliptic curves with j-invariants t, -t, 2t are all isogenous to each other.  This is predicted by the Zilber-Pink conjecture and was proved by Christopher Daw and myself.  In this talk, I will introduce the Zilber-Pink conjecture for families of elliptic curves, and outline our proof of the statement in the first sentence.  This builds on previous work of Habegger and Pila and uses transcendence properties of G-functions

Alexei Skorobogatov (Imperial College London) - 30th January 2024

Title: On p-primary torsion of the Brauer group in characteristic p

Abstract: Let k be a finitely generated field. Relation between the Tate conjecture for divisors and finiteness properties of the Brauer groups of varieties over k is well known, at least for torsion coprime to char(k). In a recent paper, D'Addezio clarified the situation for abelian varieties over fields of char(k)>0. Using similar ideas, I will show that for varieties X and Y satisfying some mild conditions, the cokernel of the natural map from the direct sum of Br(X) and Br(Y) to Br(X\times Y) is a direct sum of a finite group and a p-group of finite exponent. This implies that the transcendental Brauer group of surfaces and threefolds dominated by products of curves has finite exponent.

BAGS: Sep - Dec 2023. Organisers: Julian Lyczak and Harkaran Uppal

Samir Siksek (University of Warwick) - 12th December 2023

Title: Galois groups of low degree points on curve

Abstract: Low degree points on curves have been subject of intense study for several decades, but little attention has been paid to the Galois groups of those points. In this talk we recall primitive group actions, and focus on low degree points whose Galois group is primitive. We shall see that such points are relatively rare, and that they interfere with each other. This talk is based on joint work with Maleeha Khawaja.

Adam Morgan (University of Glasgow) - 28th November 2023

Title: Counterexamples to the Hasse principle in quadratic twist families of hyperelliptic curves

Abstract: In this talk I will discuss work in progress studying the number of counterexamples to the Hasse principle in quadratic twist families of hyperelliptic curves. In particular, I will sketch a proof that if g is at least 2, and f(x) in Z[x] is a squarefree polynomial of degree 2g+2 and Galois group S_{2g+2}, then there are infinitely many squarefree integers d for which the associated genus g hyperelliptic curve y^2=df(x) violates the Hasse principle. The argument uses results on the behaviour of Cassels—Tate pairings in quadratic twist families, in the spirit of recent work of Smith, and gives a lower bound for the number of such d that should be correct up to logarithmic factors.

Tobías Martínez (Universidad de El Salvador) - 14th November 2023

Title: The Manin Conjecture for Hirzebruch-Kleinschmidt varieties

Abstract: In this talk, we will study Manin's Conjecture on certain projective bundles over projective spaces classified by Kleinschmidt (1988). To do so, we will analyze the analytic behavior of the height zeta function induced by a metrization on the anticanonical bundle, establishing a connection between this height and the norms of hermitian vector bundles over the arithmetic curve induced by the rings of integers of the underground number field.

Alec Shute (Univeristy of Bristol) - 31st October 2023

Title: 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.

Alex Torzewski (King's College London) - 17th October 2023

Title: Studying points on varieties via varying families

Abstract: We show how Lawrence-Venkatesh's method for studying points on a variety X can be applied to curves in families. In the process, we also outline the original method. The idea is that if there exists a family over X which varies "a lot", then this strongly constrains the existence of points. This is an example of how topology influences arithmetic, in this case via p-adic differential equations!

Rachel Newton (King's College London) - 3rd October 2023

Title: Counting S4 and S5 extensions satisfying the Hasse norm principle

Abstract: Let L/K be an extension of number fields. The norm map N_{L/K}:L*->K* extends to a norm map from the ideles of L to those of K. The Hasse norm principle is said to hold for L/K if, for elements of K*, being in the image of the idelic norm map is equivalent to being the norm of an element of L*. The frequency of failure of the Hasse norm principle in families of abelian extensions is fairly well understood, thanks to previous work of Christopher Frei, Daniel Loughran and myself, as well as recent work of Peter Koymans and Nick Rome. In this talk, I will focus on the non-abelian setting and discuss joint work with Ila Varma on the statistics of the Hasse norm principle in field extensions with normal closure having Galois group S_4 or S_5.