The Spring of Rational Points

Please join us at the University of Bath on the 7th May 2024 to celebrate the visit of Jean-Louis Colliot-Thélène to the UK and the recent OBE of Roger Heath-Brown.

All talks will be in the Wolfson Lecture Theory (4 West 1.7). 

Deadline for registration was Friday 26th April. Registration is now closed.

Organiser: Daniel Loughran

Travelling to campus: https://www.bath.ac.uk/topics/travel-advice/ . Quickest option is taking the U1 bus from outside the bus station. Approx 20 min bus journey.

• 10:20 - 10:50: Arrival and coffee

• 10:50 - 11:20: Cameron Wilson (Glasgow): Diagonal quadric surfaces with a rational point

• 11:20 - 11:50: Yuan Yang (Imperial): p-primary torsion of Brauer group for two special classes of varieties

• 12:00 - 13:00: Roger Heath-Brown OBE (Oxford): Manin's conjecture for del Pezzo surfaces of degree 5 with a conic fibration

• 13:00 - 14:15: Lunch

• 14:15 - 15:15: Jean-Louis Colliot-Thélène (Paris-Saclay): Zero-cycles for rational surfaces in the function field case : an introduction to recent progress 

• 15:15 - 15:45: Coffee

• 15:45 - 16:15: Harry Shaw (Bath): Diagonal del Pezzo surfaces of degree 2 with a Brauer-Manin obstruction

• 16:15 - 16:45: Sebastian Monnet (KCL): S_n-n-ics with prescribed norms

• 17:00 - Pub and Dinner at Boho Marché

Abstracts:

Cameron Wilson (Glasgow): Diagonal quadric surfaces with a rational point

A current question of interest in the study of rational points is: given a (infinite) family of algebraic varieties, how many of them have a rational point? To answer this question one typically parameterises the family by a variety, say Y, and counts rational points on Y which correspond to rationally soluble members of your family.
After giving a brief outline on the conjectures in this area and highlighting certain results I will discuss the particular family of diagonal quadric surfaces parameterised by{Y: wx=yz}. This family was first studied by Browning, Lyczak, Sarapin who discovered that an unexpectedly large number have a rational point and attributed this odd behaviour to the presence of thin sets in Y(\mathbb{Q}) whose corresponding varieties have obvious rational points.
In upcoming work, I have shown that additional unusual behaviour is revealed once this thin set is removed from Y by providing an asymptotic for the corresponding counting problem. I will conclude with a short discussion of the main term and necessary modifications to the large sieve for quadratic characters.

Yuan Yang (Imperial): p-primary torsion of the Brauer group for two special classes of varieties

The p-primary torsion of the Brauer group of a K3 surface over an algebraically closed field of characteristic p is well-known and well-studied in the literature. There is a k-vector space part when the K3 surface is supersingular. One can see this by using flat duality theorem for surfaces and study the structure of its formal Brauer group. By using the same method, we can treat a class of similar surfaces: those surfaces with h^{2,0}=1, smooth Picard scheme, and p-torsion free Neron-Severi. Apart from this, we also study the p-primary torsion of the Brauer of ordinary varieties (ordinary in the sense of Kato) with torsion free crystalline H^2 and H^3. I will also talk about the difficulty when dealing with an arbitrary abelian 3-fold. 

Roger Heath-Brown OBE (Oxford): Manin's conjecture for del Pezzo surfaces of degree 5 with a conic fibration

A DP5 with a conic fibration may be given in 𝑃2𝑥𝑃1 by the equation

𝑦1𝑄1(𝑥1,𝑥2,𝑥3)+𝑦2𝑄2(𝑥1,𝑥2,𝑥3)=0,

with quadratic forms 𝑄1,𝑄2 defined over the rationals. The talk will discuss the proof of Manin’s conjecture for such varieties in the generic case. This is joint work with Dan Loughran.

Jean-Louis Colliot-Thélène (Paris-Saclay): Zero-cycles for rational surfaces in the function field case : an introduction to recent progress 

I shall recall the basic local-global conjecture for zero-cycles on  surfaces over global fields. In the function field case,  I shall recall how it relates to  the integral Tate conjecture for 1-cycles on certain 3-folds over a finite field.  I shall then give a short overview of  recent work of Z. Tian  and of  J. Kollár and Z. Tian. 

 Harry Shaw (Bath): Diagonal del Pezzo surfaces of degree 2 with a Brauer-Manin obstruction

 We give an asymptotic formula for the quantity of diagonal del Pezzo surfaces of degree 2 which have a Brauer-Manin obstruction to the Hasse principle when ordered by height. The proof uses similar methods as utilised by Tim Santens when considering the same problem for diagonal quartic surfaces.

Sebastian Monnet (KCL): S_n-n-ics with prescribed norms

Let α be a rational number and let Σ be a family of number fields.  For each number field K in Σ, either α is a norm of K, or it is not.  We might ask for what proportion of K in Σ this is the case.  We will see that this is a natural question to ask, and that it is extremely hard in general.  For an abelian group A, the case Σ = {A-extensions} was solved by Frei, Loughran, and Newton.  We will discuss new results for the simplest class of nonabelian extensions: so-called "generic" number fields of a given degree.