More details on the project will be provided here as they become available.
The working group will be led by Jordan Brown and Ronan O'Gorman. To participate in this working group, send an email to one of the organizers.
Both students and faculty are encouraged to participate in the working groups.
The project meets Monday - Thursday from 13:00 to 17:00 PDT in Evans 959.
Sparsity of rational points via applications.
Talks:
Lecture 1: Questions on sparsity of rational points.
Abstract: This lecture will review general conjectures of Bombieri,
Lang, and Manin describing certain geometric conditions on algebraic
varieties that should imply that their rational points are sparse. We
will give special attention to the case of subvarieties of abelian
varieties where these conjectures are proved.
Lecture 2: Bivariate injectivity on rational points.
Abstract: It is an old question of Friedman and Zagier whether there
is a bivariate polynomial f(x,y) with rational coefficients which
induces an injective function
Cornelissen
showed that this is the case under the abcd-conjecture, while Poonen
constructed examples under the Bombieri-Lang conjecture. In this talk
we will discuss Poonen's theorem and, unconditionally, we will
construct a morphism from a suitable surface (with a dense set of
rational points) to the affine line, which is injective on rational
points.
Lecture 3: Ranks and arithmetic progressions on elliptic curves.
Abstract: In the late 90's, Bremner conjectured that if an elliptic
curve over the rationals has a long sequence of rational points whose
x-coordinates are in arithmetic progression, then the elliptic curve
must have large rank. Using the uniform Mordell--Lang conjecture (now
a theorem) we will give a proof of Bremner's conjecture and outline an
extension (joint work with N. Garcia-Fritz).
Lecture 4: Manin's rational curve conjecture for certain general type surfaces.
Abstract: We will show that a well-known conjecture on the size of the
2-torsion of class groups of number fields, implies a proof of Manin's rational curve conjecture for general type surfaces with positive
irregularity. Unconditionally, we will give an upper bound on the
number of rational curves on such a surface (joint work with N. Garcia-Fritz).
References:
(1) Ooe, Top, On the Mordell--Weil rank of an abelian variety over a number field.
(2) Park, Poonen, Voight, Wood, A heuristic for boundedness of ranks of elliptic curves.
(3) Pasten, Bounded ranks and Diophantine error terms.
Working Group Project Topics
Topic 1: Ranks of abelian varieties.
(i) Develop a heuristic towards/against uniform boundedness of ranks
of abelian varieties of fixed dimension over a fixed number field.
(ii) For abelian varieties A of fixed dimension g over a fixed number
field k: prove that (or give a good heuristic for)
Topic 2: Applications of Mordell--Lang
Give new applications of the Mordell--Lang conjecture (theorem) to other Diophantine problems. For instance:
(i) Given n>1, is there a variety X over Q of dimension n with X(Q) dense in X, and a morphism f from X to the affine line, such that the induced map
is injective? This is known for n=2.
(ii) Study algebraic points of bounded degree in higher dimensional varieties (the case of curves is well-studied).
Topic 3: Manin's rational curve conjecture
Construct varieties without (or with few) rational curves and give heuristics (or proofs) for the sparsity of its rational points.