Past Seminars- Algebra

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  • Jie Xia
    Tue May 13, 2014
    2:00 pm
    Shimura varieties are defined over complex numbers and generally have number fields as the field of definition. Motivated by an example constructed by Mumford, we find conditions which guarantee a curve in char. p lifts to a Shimura curve of Hodge type. The conditions are intrinsic in positive characteristics and thus...
  • Michael J. Larsen
    Tue Apr 29, 2014
    2:00 pm
    I will discuss a number of related conjectures concerning the rational points of varieties (especially curves and abelian varieties) over fields with finitely generated Galois group and present some evidence from algebraic numebr theory, Diophantine geometry, and additive combinatorics in support of these conjectures.
  • Daniel Bertrand
    Thu Apr 24, 2014
    2:00 pm
    Contrary to their classical namesakes over the ring of integers, Pell equations over function rings in characteristic zero need not have infinitely many solutions. How often this occurs has been the theme of recent work of D. Masser and U. Zannier. The case of smooth curves is governed by the relative Manin-Mumford conjecture on abelian schemes....
  • Dimitar Jetchev
    Tue Feb 25, 2014
    4:00 pm
    We define a collection of special 1-cycles on certain Shimura 3-folds associated to U(2,1) x U(1,1) and appearing in the context of the Gan--Gross--Prasad conjectures. We study and compare the action of the Hecke algebra and the Galois group on these cycles via distribution relations and congruence relations that would ultimately lead to the...
  • Ralph Greenberg
    Tue Feb 25, 2014
    3:00 pm
    We will discuss the question of defining a p-adic L-function and formulating a main conjecture for an Artin representation. The case where the Artin representation is totally even (or odd) is classical. The corresponding main conjecture has been proven by Wiles.  This talk will discuss the special case where the representation is 2-...
  • Vladimir Baranovsky
    Thu Jun 6, 2013
    2:00 pm
  • Jennifer Balakrishnan
    Thu May 23, 2013
    2:00 pm
    We give a Chabauty-like method for finding p-adic approximations to integral points on hyperelliptic curves when the Mordell-Weil rank of the Jacobian equals the genus. The method uses an interpretation of the component at p of the p-adic height pairing in terms of iterated Coleman integrals.  This is joint work with Amnon Besser and...