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 Events
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, 3:00pm-3:50pm

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 Skolem-Mahler-Lech 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 finite-state 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 Mordell--Lang Theorem over fields of  positive characteristic. (joint with Boris Adamczewski)

Note for Attendees

Refreshments will be served between the two talks.
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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, 4:10pm-5:00pm

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 non-vanishing of quadratic twists of L-functions was somehow related to the non-vanishing of the symmetric square L-function. 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 kappa-stable 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, 3:00pm-4:10pm

Abstract

In modern language, Fermat's Descent Infini establishes that an elliptic curve has a Mordell-Weil 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 Mordell-Weil 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.

Note for Attendees

Refreshments will be served between the two talks.
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Himadri Ganguli
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, 4:10pm-5:00pm

Abstract

Please see attached PDF file.
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