Co-organizers: Gautam Chinta, Clayton Petsche, Maria Sabitova and Lucien Szpiro
This semester the seminar meets on Fridays from 4 - 5:30 PM in
The CUNY Graduate Center is located on Fifth Avenue, on the east side of the
street, between 34th and 35th Streets in midtown Manhattan.
For further information, please contact Gautam Chinta (firstname.lastname@example.org).
Title: On representations of integers in thin subgroups of SL(2,Z)
Abstract: We will talk about recent joint work with Jean
primes in the affine linear sieve.
Title: On the non-vanishing problem of theta lifts
Abstract: In this talk, after giving fairly self-contained
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
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
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.