SFU

Thu 13 Jan 2011, 3:00pm
Number Theory Seminar
Room WMAX 216 (PIMS  UBC Campus)

On vanishing coefficients of algebraic power series over fields of positive characteristic

Room WMAX 216 (PIMS  UBC Campus)
Thu 13 Jan 2011, 3:00pm3:50pm
Abstract
Let $K$ be a field of characteristic $p>0$ and let $f(t_1,\ldots ,t_d)$ be a power series in $d$ variables with coefficients in $K$. We discuss a recent generalization of both Derksen's recent analogue of the SkolemMahlerLech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices $(n_1,\ldots ,n_d)\in \mathbb{N}^d$ for which the coefficient of $t_1^{n_1}\cdots t_d^{n_d}$ in $f(t_1,\ldots ,t_d)$ is zero is generated by a finitestate automaton that accepts the base $p$ expansions of $d$tuples of natural numbers as input. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to $S$unit equations and more generally to the MordellLang Theorem over fields of positive characteristic. (joint with Boris Adamczewski)
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UBC

Thu 13 Jan 2011, 4:10pm
Number Theory Seminar
Room WMAX 216 (PIMS  UBC Campus)

The relative trace formula for SL_2

Room WMAX 216 (PIMS  UBC Campus)
Thu 13 Jan 2011, 4:10pm5:00pm
Abstract
Abstract: In a previous lecture I waved my hands at the subject of period integrals for automorphic forms on SL_2. If that lecture had a point, it was that the phenomenon of nonvanishing of quadratic twists of Lfunctions was somehow related to the nonvanishing of the symmetric square Lfunction. In this lecture, I will try to substantiate that claim by describing the relative trace formula of Jacquet, and by sketching the general shape of this formula for the group SL_2. In the end, I'll try to explain how one might get information about quadratic twists directly from the symmetric square if only one had a stable form of Jacquet's formula, and if one could explicitly evaluate the stable and kappastable portions.
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SFU

Thu 27 Jan 2011, 3:00pm
Number Theory Seminar
Room ASB 10900 (IRMACS  SFU Campus)

Explicit descent setups

Room ASB 10900 (IRMACS  SFU Campus)
Thu 27 Jan 2011, 3:00pm4:10pm
Abstract
In modern language, Fermat's Descent Infini establishes that an elliptic curve has a MordellWeil group of rank 0. Since then, the method has been generalized to provide an upper bound on the rank of any elliptic curve and further work also allows the analysis of the MordellWeil group of Jacobians of many hyperelliptic curves. Reformulating work of Schaefer, we present a general framework, in principle applicable to any curve, which allows us, under certain technical conditions, to provide an upper bound on the rank of the Jacobian of any curve. In particular, we have been able to compute some rank bounds on Jacobians of smooth plane quartic curves. This is joint work with Bjorn Poonen and Michael Stoll.
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SFU

Thu 27 Jan 2011, 4:10pm
Number Theory Seminar
Room ASB 10900 (IRMACS  SFU Campus)

On the equation f(g(x)) = f(x) h^m(x) for composite polynomials

Room ASB 10900 (IRMACS  SFU Campus)
Thu 27 Jan 2011, 4:10pm5:00pm
Abstract
Please see attached PDF file.
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Seminar Information Pages

Note for Attendees
Refreshments will be served between the two talks.