## Equivariant Tamagawa Numbers and Refined Abelian Stark Conjectures

J. Math. Sci. Univ. Tokyo
Vol. 10 (2003), No. 2, Page 225--259.

Burns, David
Equivariant Tamagawa Numbers and Refined Abelian Stark Conjectures
Let $L/K$ be a finite abelian extension of global fields, and set $G:= \Gal(L/K)$. We formulate a conjectural description of the $\Bbb Z[G]$-lattice which is generated by the leading term of the Laurent series at zero of the $G$-equivariant $L$-function associated to $L/K$ in terms of the determinant of a natural perfect complex of $\Bbb Z [G]$-modules which is defined by means of class field theory. (In the number field case this conjecture is equivalent to a special case of the Equivariant Tamagawa Number Conjecture.) We next establish a precise connection between the determinant of the specified perfect complex and suitable exterior powers of explicit unit groups. We then use this result to show that our conjecture has consequences which are closely related to certain refinements of Stark's Conjecture which have been formulated by Gross, by Tate, by Sands, by Rubin and by Popescu.