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Beyond endoscopy is the strategy put forward by Langlands for applying the trace formula to the general principle of functoriality. Subsequent papers by Langlands (one in collaboration with Frenkel and Ngo), together with more recent papers by Altug, have refined the strategy. They all emphasize the importance of understanding the elliptic terms on the geometric side of the trace formula. We shall discuss the general strategy, and how it pertains to these terms.
Beyond endoscopy is the strategy put forward by Langlands for applying the trace formula to the general principle of functoriality. Subsequent papers by Langlands (one in collaboration with Frenkel and Ngo), together with more recent papers by Altug, have refined the strategy. They all emphasize the importance of understanding the elliptic terms on the geometric side of the trace formula. We shall discuss the general strategy, and how it ...
11F66 ; 22E50 ; 22E55

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Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a random matrix model and by function field considerations.
Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a ...
11M26 ; 11M06 ; 11F66 ; 11T55 ; 11R44 ; 11M50

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This is a joint work with J. Cogdell and T.-L. Tsai. I will report on the progress made in proving the equality of Artin epsilon factors for exterior and symmetric square L-functions with those on the representation theoretic side through the local Langlands correspondence. The equality for L-functions has already been established by Henniart. I will show how the equality can be proved if one has the stability of these factors under highly ramified twists for supercuspidal representations. I will then discuss the stability question for supercuspidals by discussing how it can be deduced from a generalization of germ expansions of Jacquet and Ye from Bessel functions to certain partial Bessel functions. I will elaborate by explaining the stability in the case of GL(2) through general lemmas proved so far.
This is a joint work with J. Cogdell and T.-L. Tsai. I will report on the progress made in proving the equality of Artin epsilon factors for exterior and symmetric square L-functions with those on the representation theoretic side through the local Langlands correspondence. The equality for L-functions has already been established by Henniart. I will show how the equality can be proved if one has the stability of these factors under highly ...
11F66 ; 11F70 ; 11F80 ; 22E50