Difference between revisions of "NTS ABSTRACTSpring2020"
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−  The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of GL(n). One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and PiatetskiShapiro proved that the (NRC) fails even for the class of classical split and quasisplit groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program.  +  The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and PiatetskiShapiro proved that the (NRC) fails even for the class of classical split and quasisplit groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. 
Revision as of 18:13, 19 February 2020
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Jan 23
Rahul Krishna 
A relative trace formula comparison for the global GrossPrasad conjecture for orthogonal groups 
The global GrossPrasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a RankinSelberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases. 
Jan 30
Eric Stubley 
Class Groups, Congruences, and Cup Products 
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$part of the class group of the $p$th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$th power mod $N$. We study this rank by building off of an idea of Wake and WangErickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer. 
Feb 6
Brian Smithling 
On Shimura varieties for unitary groups 
Shimura varieties attached to unitary similitude groups are a wellstudied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic GanGrossPrasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang. 
Feb 20
Shai Evra 
Ramanujan Conjectures and Density Theorems 
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and PiatetskiShapiro proved that the (NRC) fails even for the class of classical split and quasisplit groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program.
