Spring of Rational Points II

Abstract: Given two abelian varieties over a number field K, we say that they are quadratic twists if they become isogenous after taking a quadratic extension of the base field. We moreover say that they are (strongly) locally quadratic twists if their reduction modulo almost all primes of K (or base-change to almost all completions of K) are quadratic twists. Clearly, two abelian varieties that are globally quadratic twists will also be (strongly) locally quadratic twists. The converse is not necessarily true. In this talk I will give an overview of results and counterexamples, based on joint work with E. Ambrosi and F. Fité.

Abstract: I will talk about recent progress on zero cycles on surfaces. Joint work with Ross Paterson and Sam Streeter.

Abstract: Let F be a global function field and G be an F-form of an additive group. We present a result showing that Manin's conjecture holds for a smooth equivariant compactification of G assuming appropriate conditions on the boundary. We show that any commutative unipotent group G admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and explain how this implies that the leading constant agrees with Peyre's prediction. We end with an illustrative example of projective space viewed as a compactification of an F-wound group.

There are a large selection of lunch places on campus at the University of Bath, including The Parade (fully vegan), Fountain, and Limetree. All options are ~£5-10 for lunch, or you can bring your own!

Abstract: Over a finite field k, Yanchevskiĭ asked whether a surface X is unirational when f:X->P^1_k is a conic bundle. In 1996, Mestre had supplied a positive answer when the cardinality of k is much larger than the degree of the "bad locus" of f. I will present a recent result where I answer Yanchevskiĭ's question when the "bad fibres" of f lie above rational points of P^1_k. As a bonus, and under the same conditions, the method we use proves that X has a unique R-equivalence class. These results hold more generally over quasi-finite fields. (arXiv link for the discussed paper)

Abstract: The (non-)existence of primitive integral solutions to Diophantine equations of the form Ax^p + By^q = Cz^r has been of interest to number theorists since antiquity. Interpreting such solutions as integral points on stacky curves, one may study a version of the Hasse principle for these points, and access techniques familiar from the theory of rational points on curves. In this talk, we explore how to determine whether a generalized Fermat equation of the form x^2 + By^2 = Cz^n satisfies the Hasse principle using a descent approach, generalizing results of Darmon and Granville. This work is joint with Duque-Rosero, Kobin, Roy, Sankar, and Wang.

Abstract: In this talk I will first go through the definition of the Brauer-Manin obstruction and explain what it means for a prime to potentially play a role in this obstruction. I will then explain how, under certain conditions on the reduction modulo p of the variety, primes of good ordinary reduction always potentially play a role.

Abstract: Colliot-Thélène recently asked whether every del Pezzo surface of degree 4 (dP4) has a quadratic point over a C_2 field. This question has a counterexample over C_3 fields and a positive result over C_1 fields but remained open for all C_2 fields, even in the simplest case of C((x))((y)). Note that it may be true that a dP4 always has a quadratic point over C((x))((y)), but not C((x))(y), as a dP4 always has a quadratic point over Q_p, but not Q, following from work by Creutz and Viray. We follow the method of Creutz and Viray to construct an infinite family of dP4s with a Brauer-Manin obstruction to a quadratic point over C((x))(y). This work was funded by a Heilbronn Focused Research Grant, and is joint work with Giorgio Navone, Harry Shaw and Haowen Zhang. 

Social dinner at Franco Manca Bath. Franco Manca is right next to the train station, so it should be easy for people with trains to catch, and we will probably head to the pub afterwards for anyone who wants to join.