Collaborative Number Theory Seminar

Title: On representations of integers in thin subgroups of SL(2,Z)

Abstract: We will talk about recent joint work with Jean Bourgain, obtaining
primes in the affine linear sieve.

Title: On the non-vanishing problem of theta lifts

Abstract: In this talk, after giving fairly self-contained backgrounds
on theta lifting, we will discuss some recent results on non-vanishing
of global theta lifts. This is partly a joint work with Wee Teck Gan.

(Prof. Sabitova's talk is cancelled due to illness)

Title: A Diophantine Frobenius Problem: the Largest Non-Genus of a Cyclic Group

Abstract: We obtain sharp upper and lower bounds on a certain
four-dimensional Frobenius number determined by a prime pair (p,q),
including exact formulae for two infinite subclasses of such pairs.
The problem is motivated by the study of compact (Riemann) surfaces
which are regular pq-fold coverings of surfaces of lower genus. In
this context, the Frobenius number is (up to an additive translation) the
largest genus in which no surface is such a covering. The general n-dimesnional
Frobenius problem ($n \geq 3$) is NP-hard, and it is not clear whether
our restricted problem retains this property. Our methods are elementary:
only some linear algebra, the division algorithm, and inequalities.
This is joint work with Cormac O' Sullivan.

Title: D-ratio and its applications

Abstract: When f: P^n -> P^n be a morphism of degree d, then we have following inequality:
1/d h( f(P) ) - C_1 < h( P ) < 1/d h ( f(P) ) + C_2.
If f: P^n -> P^n is a rational map, then the second inequality
h( P ) < 1/d h ( f(P) ) + C_2
is invalid because of the failure of the functorial property of Weil Height
Machine. However, by defining new invariant of rational map which is called
D-ratio, we can get similar inequality:
1/d h( f(P) ) - C_1 < h( P ) < r(f) /d h ( f(P) ) + C_2
where r(f) is the D-ratio of f. This inequality will give us some applications
in Arithmetic Dynamics - the height boundedness of preperiodic points of a rational map f.
In addition, D-ratio can provides another applications - proof of Kawaguchi's
Conjecture, improvement of Silverman's result for jointly regular pair of rational maps, etc.

Title: Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group

Abstract: Let E be an elliptic surface over a curve C, defined over a
number field K. A reasonable question to ask is "To what extent does
the geometry of the fibration dictate the arithmetic of the fibres?"
Specializing at a fibre gives a homomorphism from the group of
sections E(C) of E to the Mordell-Weil group E_t(K) of the fibre. I
will discuss the properties of this specialization, with an emphasis
on the size of the torsion subgroup of E_t(K) modulo the image of
E(C), which one might think of as a measure of surjectivity of the
specialization that takes ignores differences in rank.

Title: On the minimum canonical height for an elliptic curve over C(t)

Lang's height conjecture postulates a uniform lower bound for the
canonical height of a nontorsion point on an elliptic curve. Hindry and
Silverman proved Lang's height conjecture under the hypothesis of a
conjecture of Lucien Szpiro. Szpiro's conjecture is equivalent to the ABC
conjecture, which is known over function fields. It is natural to ask:
What is the smallest possible canonical height of a nontorsion point
on an elliptic curve over a function field $K$? We find new record
for the canonical height of a nontorsion point on an elliptic curve over
C(t), and give heuristics suggesting this is the minimum possible.