Difference between revisions of "NTS ABSTRACT"
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The epipelagic representations of Reeder-Yu, a generalization of the "simple supercuspidals" of Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups G. Given a "stable functional" f, which is a suitably 'generic' linear functional on a vector space coming from a Moy-Prasad filtration for G, one can create such a representation. It is known that the representations created in this way are compactly induced from the fixer in G of f and it is important to identify explicitly all the elements that belong to this fixer. This work is in-progress. | The epipelagic representations of Reeder-Yu, a generalization of the "simple supercuspidals" of Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups G. Given a "stable functional" f, which is a suitably 'generic' linear functional on a vector space coming from a Moy-Prasad filtration for G, one can create such a representation. It is known that the representations created in this way are compactly induced from the fixer in G of f and it is important to identify explicitly all the elements that belong to this fixer. This work is in-progress. | ||
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+ | == Sep 17 == | ||
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+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Zureick-Brown''' | ||
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+ | | bgcolor="#BCD2EE" align="center" | ''Tropical geometry and uniformity of rational points'' | ||
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+ | Let X be a curve of genus g over a number field F of degree d = [F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known to follow from the Bomberi-Lang conjecture. We prove a special case of this conjecture - we give an explicit uniform bound when X has Mordell-Weil rank r ≤ g-3. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel-Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate. Our methods combine Chabauty-Coleman's p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs. This is joint work with Joe Rabinoff and Eric Katz. | ||
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Revision as of 13:40, 31 August 2015
Return to NTS Fall 2015
Sep 03
Kiran Kedlaya |
On the algebraicity of (generalized) power series |
A remarkable theorem of Christol from 1979 gives a criterion for detecting whether a power series over a finite field of characteristic p represents an algebraic function: this happens if and only if the coefficient of the n-th power of the series variable can be extracted from the base-p expansion of n using a finite automaton. We will describe a result that extends this result in two directions: we allow an arbitrary field of characteristic p, and we allow "generalized power series" in the sense of Hahn-Mal'cev-Neumann. In particular, this gives a concrete description of an algebraic closure of a rational function field in characteristic p (and corrects a mistake in my previous attempt to give this description some 15 years ago). |
Sep 10
Sean Rostami |
Fixers of Stable Functionals |
The epipelagic representations of Reeder-Yu, a generalization of the "simple supercuspidals" of Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups G. Given a "stable functional" f, which is a suitably 'generic' linear functional on a vector space coming from a Moy-Prasad filtration for G, one can create such a representation. It is known that the representations created in this way are compactly induced from the fixer in G of f and it is important to identify explicitly all the elements that belong to this fixer. This work is in-progress. |
Sep 17
David Zureick-Brown |
Tropical geometry and uniformity of rational points |
Let X be a curve of genus g over a number field F of degree d = [F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known to follow from the Bomberi-Lang conjecture. We prove a special case of this conjecture - we give an explicit uniform bound when X has Mordell-Weil rank r ≤ g-3. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel-Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate. Our methods combine Chabauty-Coleman's p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs. This is joint work with Joe Rabinoff and Eric Katz. |
Oct 15
Valentin Blomer |
Coming soon... |
Coming soon... |
Dec 17
Nathan Kaplan |
Coming soon... |
Coming soon... |