Difference between revisions of "NTS Spring 2014/Abstracts"
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Abstract: While we know by Faltings' theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of Sym^d(X) that are not parametrized by a projective space or a coset of an abelian variety, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d. | Abstract: While we know by Faltings' theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of Sym^d(X) that are not parametrized by a projective space or a coset of an abelian variety, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d. | ||
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+ | == March 11 == | ||
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+ | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
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+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yueke Hu''' (Madison) | ||
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+ | | bgcolor="#BCD2EE" align="center" | Title: Local integrals of triple product ''L''-function and subconvexity bound | ||
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+ | | bgcolor="#BCD2EE" | | ||
+ | Abstract: Venkatesh proposed a strategy to prove the subconvexity bound in the level aspect for triple product ''L''-function. With the integral representation of triple product ''L''-function, if one can get an upper bound for the global integral and a lower bound for the local integrals, then one can get an upper bound for the ''L''-function, which turns out to be a subconvexity bound. Such a subconvexity bound was obtained essentially for representations of square free level. I will talk about how to generalize this result to the case with higher ramifications as well as joint ramifications. | ||
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Revision as of 16:52, 10 March 2014
Contents
January 23
Majid Hadian-Jazi (UIC) |
Title: On a motivic method in Diophantine geometry |
Abstract: By studying the variation of motivic path torsors associated to a variety, we show how certain nondensity assertions in Diophantine geometry can be translated to problems concerning K-groups. Then we use some vanishing theorems to obtain concrete results. |
January 30
Alexander Fish (University of Sydney, Australia) |
Title: Ergodic Plunnecke inequalities with applications to sumsets of infinite sets in countable abelian groups |
Abstract: By use of recent ideas of Petridis, we extend Plunnecke inequalities for sumsets of finite sets in abelian groups to the setting of measure-preserving systems. The main difference in the new setting is that instead of a finite set of translates we have an analogous inequality for a countable set of translates. As an application, by use of Furstenberg correspondence principle, we obtain new Plunnecke type inequalities for lower and upper Banach density in countable abelian groups. Joint work with Michael Bjorklund, Chalmers. |
February 13
John Voight (Dartmouth) |
Title: Numerical calculation of three-point branched covers of the projective line |
Abstract: We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups. This is joint work with Michael Klug, Michael Musty, and Sam Schiavone. |
February 20
Nir Avni (Northwestern) |
Title: Representation zeta functions |
Abstract: I will talk about connections between the following: 1) Asymptotic representation theory of an arithmetic lattice G(Z). More precisely, the question of how many n-dimensional representations does G(Z) have. 2) The distribution of a random commutator in the p-adic analytic group G(Z_{p}). 3) The complex geometry of the moduli spaces of G-local systems on a Riemann surface, and, more precisely, the structure of its singularities. |
February 27
Jennifer Park (MIT) |
Title: Effective Chabauty for symmetric power of curves |
Abstract: While we know by Faltings' theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of Sym^d(X) that are not parametrized by a projective space or a coset of an abelian variety, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d. |
March 11
Yueke Hu (Madison) |
Title: Local integrals of triple product L-function and subconvexity bound |
Abstract: Venkatesh proposed a strategy to prove the subconvexity bound in the level aspect for triple product L-function. With the integral representation of triple product L-function, if one can get an upper bound for the global integral and a lower bound for the local integrals, then one can get an upper bound for the L-function, which turns out to be a subconvexity bound. Such a subconvexity bound was obtained essentially for representations of square free level. I will talk about how to generalize this result to the case with higher ramifications as well as joint ramifications. |
Organizer contact information
Sean Rostami
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