Stark's conjecture

Stark's conjecture (or conjectures) in number theory, introduced in a series of papers in the 1970s by American mathematician Harold Stark, deals partly with the question of finding interesting, particular units in number fields. This has resulted in a large conjectural development for "L"-functions, and is also capable of producing concrete, numerical results, as they are precise enough for computation.

Formulation

Stark's principal conjecture proposes that the first non-zero coefficient of the Taylor expansion of an "L"-function at zero is given by the product of an algebraic number and the so-called Stark regulator. When the extension defining the "L"-function is abelian, various refined conjectures have been proposed (see the paper by Rubin [ [http://www.numdam.org/item?id=AIF_1996__46_1_33_0 Rubin: A Stark conjecture “over ${f Z}$” for abelian $L$-functions with multiple zeros ] ] , for example), which in certain cases predict the existence of special units known as Stark units.

Computation

The first order zero conjectures are used in recent versions of the PARI/GP computer algebra system to compute Hilbert Class fields of totally real number fields, and the conjectures provide one solution to Hilbert's twelfth problem, which challenged mathematicians to show how class fields may be constructed over any number field by the methods of complex analysis.

Stark's principal conjecture has been proven in various special cases, including the case where the character defining the "L"-function takes on only rational values. Except when the base field is the rational numbers or an imaginary quadratic field, the abelian Stark conjectures are still unproved in number fields, and more progress has been made in function fields.

Progress

Work of Manin related Stark's conjectures to noncommutative geometry of Alain Connes. This provides a very attractive conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin's techniques will yield the actual proof.

References

* [http://www.math.umass.edu/~dhayes/lecs.html Homepage of David R. Hayes]
* [http://front.math.ucdavis.edu/math.AG/0202109 Manin's paper]
* [http://www.numdam.org/item?id=AIF_1996__46_1_33_0 Rubin's paper]