Joint IAS/PU Number Theory
A continuous representation of a profinite group induces a continuous pseudorepresentation, where a pseudorepresentation is the data of the
characteristic polynomial coefficients. We discuss the geometry of the resulting map from the moduli formal groupoid of representations to the moduli formal scheme of pseudorepresentations.
The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite. This talk will describe an approach to the converse. The methods are mostly p-adic.
A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. This is a powerful local-to-global principle which is very suitable for analytic and arithmetic applications. One can formulate a Whittaker analogue of the Ichino-Ikeda conjecture. We use the descent method of Ginzburg-Rallis-Soudry to reduce the Whittaker version to a purely local identity which we prove in the p-adic case under some mild hypotheses. Joint work with Zhengyu Mao
The trace formula has been the most powerful and mainstream tool in automorphic forms for proving instances of Langlands functoriality, including character relations. Its generalization, the relative trace formula, has also been used to prove functoriality between "spaces" (more precisely: spherical homogeneous varieties), which is a generalization of functoriality for groups, including relations between periods of automorphic forms.
Application of Plancherel's theorem to integral kernels approximating compact period functionals yields estimates on (global) automorphic Levi-Sobolev norms of the functionals.
Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in terms of a number field K and the dimension of A). We achieve this by considering abelian varieties in families and then using a special variant of Hilbert's irreducibility theorem. Some results of Serre on the mod ell Galois image will also be reviewed. (This is joint work with David Zureick-Brown)
We study open-closed orbifold Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This is a joint work with Bohan Fang and Hsian-Hua Tseng.
We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field.
Motivated by work of Kato and others for n=3, we show that local-global principles hold for
H^n(F, Z/mZ(n-1)) for all n>1.
In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for H^1(F,G), a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G.