3:00–4:00 pm
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My Favorite Algebraic Cycle
Notes for this talk can be found here.
Abstract:
Hasse Weil L-functions L(H^n(X),s) are functions of a complex variable s, analytic in a half-plane, associated to an algebraic variety X defined over Q (or more generally any number field), together with a choice of cohomological degree n. They have an Euler product expansion with local factors at good reduction primes p of the form (f_p = geometric frobenius at p)
1/det(1-f_pp^{-s}|H^n(X_{\F_p})).
The simplest is the Riemann zeta function. Another example is L(H^1(E),s) with E an elliptic curve. Significant progress has been made in understanding these in recent years due to the introduction of Euler systems by Kolyvagin and Kato.
The basic conjecture (first formulated by Birch and Swinnerton-Dyer, and then greatly generalized by Beilinson) is that the behavior of L at integer values of s should be controlled by algebraic cycles.
I will focus on the case dim X = 2m-1 is odd, and n=2m-1. The order of vanishing of L(H^{2m-1}(X),s) at s = m is then conjectured to be the rank of the Chow group of codim m algebraic cycles homologous to 0 modulo rational equivalence on X, CH^m(X)^0. For example, when dim X=1, this becomes the Mordell-Weil group of 0-cycles on X of degree 0 modulo divisors of functions. The order of 0 at s=1 should be the rank of the Mordell-Weil group, and the value of the first non-vanishing term in the Taylor series expansion of L(H^1(X),s) in s at s=1 should be given upto elementary factors by the discriminant of the height pairing on the Mordell Weil group.
Not much is known in the case dim X=2m-1>1. The talk will focus on a class of elementary examples of varieties X and cycles Z. The construction is geometric rather than arithmetic, and there are interesting links with hypergeometric motives and limiting mixed Hodge structures.