An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a beautiful product formula, purely in terms of congruence considerations involving that polynomial, for the size of such an isogeny class; an equidistribution hypothesis too strong to be true apparently calculates this cardinality!
I will give a new, transparent explanation, worked out with Julia Gordon, for this phenomenon. It turns out that Gekeler's formula computes an adelic orbital integral which, thanks to work of Langlands and Kottwitz, visibly calculates the desired quantity.
Local heuristics and exact formulas for elliptic curves over finite fields
Feb. 24, 2015 3pm (Math 220)
Functional Analysis
Ian Long (University of Colorado, Boulder)
X
A finite measure on the real line is said to be spectral if there exists a set of exponential functions which forms an orthonormal basis for . Moreover, a bounded Lebesgue measurable set of positive measure is spectral if the renormalization of Lebesgue measure on is spectral. I will introduce examples of spectral measures and sets on the real line, describe their relation to an unproven conjecture of Fuglede, examine a related conjecture made by Gabardo and Lai, and introduce a recent result by Duktay and Jorgensen which attempts to answer the question: when do two spectral measures
Spectral Measures Sponsored by the Meyer Fund
Feb. 24, 2015 3pm (Math 350)
Algebraic Geometry
Zheng Zhang (Stony Brook)
X
Based on the work of Gross and Sheng-Zuo, Friedman and Laza have classified variations of real Hodge structure of Calabi-Yau type over Hermitian symmetric domains. In particular, over every irreducible Hermitian symmetric domain there exists a canonical variation of real Hodge structure of Calabi-Yau type. In this talk, we will first review Friedman and Laza’s classification. A natural question to ask is whether the canonical Hermitian variations of Hodge structure of Calabi-Yau type come from families of Calabi-Yau manifolds (geometric realization). In general, this is very difficult and is still open for small dimensional domains. We will discuss an intermediate question, namely does the canonical variations occur in algebraic geometry as sub-variations of Hodge structure of those coming from families of algebraic varieties (motivic realization). In particular, we will give motivic realizations for the canonical variations of Calabi-Yau type over irreducible tube domains of type A using abelian varieties of Weil type.
Local heuristics and exact formulas for elliptic curves over finite fields