Planning
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Friday, December 9, 2022 |
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09:00
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17:00
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›10:45 (1h)
Claire Burrin: Windings of prime geodesics via the trace formula
A standard construction in automorphic forms associates a multiplier system to any nonuniform Fuchsian group. I will explain how this construction extends to give a notion of winding number for closed geodesics on the linear space of the corresponding hyperbolic surface. I will then show a Dirichlet-type equidistribution theorem for various families of cusped hyperbolic surfaces: For every set A of integers with density d=d(A), the set of prime geodesics with winding number in A has density 1/d. This is joint work with Flemming von Essen.
› B405
›12:00 (1h45)
› Faculty restaurant
›13:45 (1h)
Asbjørn Nordentoft: The distribution of closed geodesics in the homology of modular curves
Given an element of the (wide) class group of a real quadratic field K, one can associate an (oriented) closed geodesic on the modular curve of square-free level N (when all primes dividing N splits in K). Duke proved that as the discriminant tends to infinity the (oriented) closed geodesics equidistribute. In this talk we will discuss the related problem regarding the distribution of the homology classes associated to the closed geodesics. Our main result is a convergence (in an appropriate sense that we will explain) to the Eisenstein element in the homology. This is the real quadratic analogue of the equidistribution of supersingular reduction of CM elliptic curves (as studied by Michel , Liu—Masri—Young, …). If time permits we will discuss a similar result for p-orbits of (oriented) closed geodesics (i.e. packets of closed geodesics associated to the units modulo p), as well as the connection to joint work with Peter Humphries on orbifolds associated to real quadratic fields.
› B405
›15:15 (1h)
Peter Humphries: Sparse equidistribution of hyperbolic orbifolds
Duke, Imamoḡlu, and Tóth have recently constructed a new geometric invariant, a hyperbolic orbifold, associated to each narrow ideal class of a real quadratic field. Furthermore, they have shown that the projection of these hyperbolic orbifolds onto the modular surface equidistributes on average over a genus of the narrow class group as the fundamental discriminant of the real quadratic field tends to infinity. We discuss a refinement of this result, sparse equidistribution, and connect this to cycle integrals of automorphic forms and subconvexity for Rankin-Selberg L-functions. This is joint work with Asbjørn Nordentoft.
› B405
›16:30 (1h)
Maksym Radziwiłł: Non vanishing of twists of GL(4) L-functions
I will discuss recent work with Lyiang Yang in which we establish the non vanishing of L(1/2, \pi \times \chi) for infinitely many primitive Dirichlet characters and where \pi is either a GL(4) cuspidal automorphic representation of cohomological type or arises from a functorial lift e.g Sym^3 g with g a GL(2) cuspidal automorphic representation. The exact family of \pi's for which our result works is quite a bit larger. I will describe the exact conditions under which the result holds.
› B405
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B405
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