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).

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.