In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields.
Stark units in the abelian rank-one case have been computed in specific examples, allowing verification of the veracity of his refined conjecture.
Stark's principal conjecture has been proven in a few special cases, such as when the character defining the L-function takes on only rational values.
[2] This provides a conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin's techniques will yield the actual proof.
[10] In 1999, Cristian Dumitru Popescu proposed a function field analogue of Rubin's conjecture and proved it in some cases.