Co-organizers: Gautam Chinta, Clayton Petsche, Maria Sabitova and Lucien Szpiro
This semester the seminar meets on Fridays from 4 - 5:30 PM in Room 4422. 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 (chinta@sci.ccny.cuny.edu). For information on the seminar in previous years, please see this link.
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: TBA