Andreas Malmendier (USU)
Title: Fourier coefficients and L-values for modular forms on G_2
Abstract: Inose surfaces, associated with Kummer surfaces of abelian surfaces, provide a purely geometric interpretation for a certain duality in string theory. The reason is that they can be described in various ways: as quartic projective hypersurfaces, double-sextic surfaces, and elliptically fibered K3 surfaces. Building on this foundation and generalizing to lower Picard rank, in recent work we obtained an explicit description for certain two-elementary K3 surfaces with finite automorphism group of Picard rank greater than 12, associated with the double covers of the projective plane branched along certain sextics, together with a description of their inverse period map. We then determine explicit generators for the ring of modular forms associated with the moduli spaces of the K3 surfaces, and show that the modular form generators appear as coefficients in the Weierstrass-type equations describing a canonical Jacobian elliptic fibration. This is joint work with Adrian Clingher and Xavier Roulleau, and with Adrian Clingher and Brandon Williams.
Matthew Bertucci (U of U)
Title: Generalized Bertini Theorems over Finite Fields
Abstract: The classical Bertini theorem says that a general hyperplane section of a smooth projective variety is also smooth. This is not true over finite fields, but Poonen showed there exist smooth hypersurface sections in sufficiently high degree, and these have positive density as the degree of the hypersurface goes to infinity. His result also allows prescribing the first few Taylor coefficients of sections at finitely many points. In the motivic setting, Bilu and Howe proved an analogous result but allowing much more general Taylor conditions. We extend Poonen's result in the arithmetic setting to Taylor conditions arising as subsheaves of the sheaf of differentials such that the corresponding quotient is locally free.
Petar Bakic (U of U)
Title: Fourier coefficients and L-values for modular forms on G_2
Abstract: In his classic work on the Shimura correspondence, Waldspurger used the theta correspondence to relate the Fourier coefficients of certain modular forms of half-integral weight to special values of the corresponding L-functions.
In this talk, we investigate an analogous formula for Fourier coefficients of modular forms on the group G_2, using the exceptional theta lift from PU(3). This is the subject of a long-standing conjecture of Gross.
This is work in progress, joint with A. Horawa, S.D. Li-Huerta, and N. Sweeting.
Mitchell Pound (USU)
Title: Towards generalized abelianizations of symplectic quotients
Abstract: Many spaces in geometry and physics arise as quotients of a given manifold by a group action. In symplectic geometry, the Marsden–Weinstein quotient (or symplectic reduction) constructs such quotients for Hamiltonian group actions, producing new symplectic or stratified symplectic spaces. In a recent paper, Crooks and Weitsman showed that the symplectic quotient by a compact connected Lie group can also be realized as a symplectic quotient by a torus, a process known as abelianization. I will discuss the intuition and implications of these abelianization results, as well as attempts at a generalization.
Daniel Gulotta (U of U)
Title: Generalizing local Jacquet-Langlands via Hecke correspondences
Abstract: Langlands functoriality predicts that reductive groups that are inner forms of each other, such as the group GL_2(R) of invertible 2x2 real matrices, and the group H^{\times} of nonzero quaternions, have similar representation theory.
One can compare representations of the groups in the following ways:
- Each representation has a "Harish-Chandra character", which is a generalization of the notion of trace for certain infinite-dimensional representations. One can then look for identities relating Harish-Chandra characters.
- One can find a space that has an action of both groups, and look at how the groups act on its cohomology.
In the case of reductive groups over a p-adic field, Hansen-Kaletha-Weinstein have used group actions on the cohomology of "local shtuka spaces" to prove identities relating Harish-Chandra characters. These identities only cover the elliptic locus of each group. I will describe some improvements to their method that allow identities to be extended beyond the elliptic locus.
While my results only apply to groups over p-adic fields, I believe the ideas could also be useful in the case of groups over the real numbers. I will give an idea of the objects of interest in the real case.