CUNY Graduate Center

Collaborative Number Theory Seminar

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 (

Schedule, Fall 2009

  • October 9, Alex Kontorovich (Brown University/IAS)

    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.

  • October 16, no seminar

  • October 23, Shuichiro Takeda (Purdue University)

    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.

  • October 30, No seminar

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

  • November 6, Anthony Weaver (Bronx Community College, CUNY)

    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.

  • November 13, ChongGyu Lee (Brown University)

    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.

  • November 20, Patrick Ingram (University of Waterloo)

    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.

  • November 27, no seminar
  • December 4, Sonal Jain (NYU)

    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.