Bay Area Algebraic Number Theory and Arithmetic Geometry Day 10
Saturday, February 21, 2015
Braun Auditorium, Mudd Building (Chemistry Department)
Mirela Ciperiani, University of Texas, Austin
Luis Garcia, Imperial College
Wei Ho, University of Michigan
Daniel Litt, Stanford University
Ken Ono, Emory University
Please RSVP to sdasgup2 (at) ucsc (dot) edu
Titles and Abstracts:
Mirela Ciperiani, "Local points of supersingular elliptic curves on Z_p-extensions"
Work of Kobayashi and Iovita-Pollack describes how local points of supersingular elliptic curves on ramified Zp-extensions of Qp split into two strands of even and odd points. We will discuss a generalization of this result to Zp-extensions that are localizations of anticyclotomic Zp-extensions over which the elliptic curve has non-trivial CM points.
Luis Garcia, "Theta lifts and currents on Shimura varieties"
The Shimura varieties X attached to orthogonal and unitary groups come equipped with a large collection of so-called special cycles. Examples include Heegner divisors on modular curves and Hirzebruch-Zagier cycles on Hilbert modular surfaces. We will review work of Borcherds and Bruinier using regularised theta lifts for the pair (SL2,O(V)) to construct Green currents for special divisors. Then we will explain how to construct other interesting currents on X using the dual pair (Sp4,O(V)). We will show that one obtains currents in the image of the regulator map of a certain motivic complex of X. Finally, we will describe how an argument using the Siegel-Weil formula allows to relate the values of these currents to the product of a special value of an L-function and a period on a certain subgroup of Sp4.
Wei Ho, "Distributions of ranks of elliptic curves"
In the last five years, there has been significant theoretical progress on understanding the average rank of all elliptic curves over Q, ordered by height, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as discuss generalizations in many directions (e.g., to other families of elliptic curves, higher genus curves, and higher-dimensional varieties) and some corollaries of these types of theorems. We will also describe recently collected data on ranks and Selmer groups of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt).
Daniel Litt, "Geometric Lefschetz Hyperplane Theorems"
Classical results of Lefschetz and Grothendieck compare the topology of a smooth projective variety X over the complex numbers to that of an ample divisor D on X. For example, if the dimension of X is at least 4, the canonical map from Pic(X) to Pic(D) is an isomorphism, and if the dimension of X is at least 3, the canonical map on etale π1 from D to X is an isomorphism.
This talk will compare the geometry and arithmetic of a variety X to that of an ample divisor D on X, from a functor-of-points viewpoint. That is, we will provide general criteria for the map from Hom(X, S) to Hom(D, S) to be an isomorphism, where S is a scheme or stack. Taking S to be particular spaces (e.g. BG for G a finite etale group scheme) recovers various classical Lefschetz theorems (in this case, Lefschetz for etale π1), and many new ones. As an example application, we show that if X is smooth and projective of dimension at least 3 over a characteristic zero field, D
is an ample divisor on X, and f: C ---> X is a smooth projective curve of genus at least 1, then the restriction map from Sections(f) to
Sections(f_D) is an isomorphism. Though this result is for characteristic zero, the proof goes through characteristic p.
Ken Ono, "Moonshine"
In the early 1970s it was noticed that the prime divisors of the order of the Monster (which was not yet proven to exist), the largest sporadic finite simple group,
are the levels appearing in Ogg's classification of hyperelliptic modular curves. Ogg offered a bottle of Jack Daniels for a good explanation of this strange coincidence.
Ten years later McKay noticed that 196884=196883+1, where 196884 is the first nontrivial coefficient of the j-function, and 196883 and 1 are the dimensions of
the two smallest irreducible representations of the Monster. McKay's observation and Ogg's problem are the first hints of the Conway-Norton Monstrous Moonshine Conjecture,
which was proved by Borcherds. In 2010 three Japanese physicists observed that the coefficients of a certain specialization of the K3 elliptic genus could similarly
be expressed in terms of the small dimensions of the irreducible representations of the Mathieu group M24. That work has inspired much recent activity at the
interface of mathematical physics and number theory. It gave birth to the Umbral Moonshine Conjecture. In this lecture the speaker will review the history of Moonshine,
and he will describe the recent work in the area including the proof of the Umbral Moonshine Conjecture. This is joint work with John Duncan and Michael Griffin.
Parking is free and plentiful on Roth Way and in the Oval on weekends. Click here for a campus parking map. The Mudd Building is at E7.
There is no formal registration, but if you plan to attend, we would appreciate an email to sdasgup2 at ucsc dot edu to
help plan the event, especially if you plan to attend the dinner afterwards.
There will be a dinner following the conference at 6:00pm, at Reposado (236 Hamilton Ave, Palo Alto).
Please send an email to sdasgup2 at ucsc dot edu if you plan to attend. Graduate students will be partially subsidized at the dinner. We thank the UC Santa Cruz Mathematics Department, the Math Research Center at Stanford University, and the UC Berkeley Mathematics Department for partial financial support.