David Lilienfeldt: Elements in K4 and regulator maps of Fermat curves

Beilinson's conjectures predict that special values of L-functions of varieties are given, up to rational factors, by regulators of elements in algebraic K-theory. In this talk, I will present a uniform construction of non-trivial elements in K4 of the Fermat curves of degree N using polylogarithmic complexes. Their regulators have explicit formulas involving Zagier's trilogarithm and hypergeometric functions, and they grow like 3⁄2 zeta(s)N^2. When N=3,4,6, I will provide numerical verifications of Beilinson's conjecture for L-values at s=3.

This is joint work with François Brunault and Yusuke Nemoto.