ucsd number theory seminar (math 209)

october 2

organizational meeting

october 16

jon aycock (uc san diego)
jacobians of graphs via edges and iwasawa theory

the jacobian (or sandpile group) is an algebraic invariant of a graph that plays a similar role to the class group in classical number theory. there are multiple recent results controlling the sizes of these groups in galois towers of graphs that mimic the classical results in iwasawa theory, though the connection to the values of the ihara zeta function often requires some adjustment. in this talk we will give a new way to view the jacobian of a graph that more directly centers the edges of the graph, construct a module over the relevant iwasawa algebra that nearly corresponds to the interpolated zeta function, and discuss where the discrepancy comes from.

october 23
+pre-talk

jennifer johnson-leung (university of idaho)
rationality of certain power series attached to paramodular siegel modular forms

the euler product expression of the dirichlet series of fourier coefficients of an elliptic modular eigenform follows from a formal identity in the hecke algebra for gl(2) with full level. in the case of siegel modular forms of degree two with paramodular level, the situation is more delicate. in this talk, i will present two rationality results. the first concerns the dirichlet series of radial fourier coefficients for an eigenform of paramodular level divisible by the square of a prime. this result is an application of the theory of stable klingen vectors (joint work with brooks roberts and ralf schmidt). while we are able to calculate the action of certain hecke operators on eigenforms, the structure of the hecke algebra of deep level is not known in general. however, in the case of prime level, there is a robust description of the local hecke algebra which yields a rationality result for a formal power series of hecke operators (joint work with joshua parker and brooks roberts). in both cases, we obtain the expected local l-factor as the denominator of the rational function.

october 30
+pre-talk

chengyang bao (ucla)
computing crystalline deformation rings via the taylor-wiles-kisin patching method

crystalline deformation rings play an important role in kisin's proof of the fontaine-mazur conjecture for gl2 in most cases. one crucial step in the proof is to prove the breuil-mezard conjecture on the hilbert-samuel multiplicity of the special fiber of the crystalline deformation ring. in pursuit of formulating a horizontal version of the breuil-mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. our approach, based on reverse-engineering the taylor-wiles-kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.

november 6
+pre-talk

brandon alberts (eastern michigan)
inductive methods for counting number fields

we will discuss an inductive approach to determining the asymptotic number of g-extensions of a number field with bounded discriminant, and outline the proof of malle's conjecture in numerous new cases. this talk will include discussions of several examples demonstrating the method.

november 13

chris xu (uc san diego)
special cycles on $g_2$

on the symmetric space for $g_2$, there exist various submanifolds $g_d$ corresponding to the stabilizer of a norm $d$ vector. we show that when a suitable automorphic form is integrated against the $g_d$, the resulting numbers assemble to give a half-integral weight classical modular form. although this is already implied by a result of kudla-millson, we give a simpler proof that avoids the complications in their paper.

november 20

linli shi (connecticut)
on higher regulators of picard modular surfaces

the birch and swinnerton-dyer conjecture relates the leading coefficient of the l-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. beilinson’s conjectures generalize the bsd conjecture to formulas for values of motivic l-functions at non-critical points. in this talk, i will relate motivic cohomology classes, with non-trivial coefficients, of picard modular surfaces to a non-critical value of the motivic l-function of certain automorphic representations of the group gu(2,1).

december 4
+pre-talk

yu fu (caltech)
the p-adic analog of the hecke orbit conjecture and density theorems toward the p-adic monodromy

the hecke orbit conjecture predicts that hecke symmetries characterize the central foliation on shimura varieties over an algebraically closed field $k$ of characteristic $p$. the conjecture predicts that on the mod $p$ reduction of a shimura variety, any prime-to-p hecke orbit is dense in the central leaf containing it, and was recently proved by a series of nice papers. however, the behavior of hecke correspondences induced by isogenies between abelian varieties in characteristic $p$ and $p$-adically is significantly different from the behavior in characteristic zero and under the topology induced by archimedean valuations. in this talk, we will formulate a $p$-adic analog of the hecke orbit conjecture and investigate the $p$-adic monodromy of $p$-adic galois representations attached to points of shimura varieties of hodge type. we prove a density theorem for the locus of formal neighborhood associated to the mod $p$ points of the shimura variety whose monodromy is large and use it to deduce the non-where density of hecke orbits under certain circumstances.