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. The conjectures generalize the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function of a number field as the product of a regulator related to S-units of the field and a rational number. When K/k is an abelian extension and the order of vanishing of the L-function at s = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units, which generate abelian extensions of number fields.
Formulation
General case The Stark conjectures, in the most general form, predict that the leading coefficient of an Artin L-function is the product of a type of regulator, the Stark regulator, with an algebraic number.
Abelian rank-one case When the extension is abelian and the order of vanishing of an L-function at s = 0 is one, Stark's refined conjecture predicts the existence of Stark units, whose roots generate Kummer extensions of K that are abelian over the base field k (and not just abelian over K, as Kummer theory implies). As such, this refinement of his conjecture has theoretical implications for solving Hilbert's twelfth problem.
Computation Stark units in the abelian rank-one case have been computed in specific examples, allowing verification of the veracity of his refined conjecture. These also provide an important computational tool for generating abelian extensions of number fields, forming the basis for some standard algorithms for computing abelian extensions of number fields. The first rank-zero cases 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.
Progress 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. Except when the base field is the field of rational numbers or an imaginary quadratic field, which were covered in the work of Stark, the abelian Stark conjectures is still unproved for number fields. More progress has been made in function fields of an algebraic variety. Manin (2004) related Stark's conjectures to the noncommutative geometry of Alain Connes. 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.
Variations In 1980, Benedict Gross formulated the Gross–Stark conjecture, a p-adic analogue of the Stark conjectures relating derivatives of Deligne–Ribet p-adic L-functions (for totally even characters of totally real number fields) to p-units. This was proved conditionally by Henri Darmon, Samit Dasgupta, and Robert Pollack in 2011. The proof was completed and made unconditional by Dasgupta, Mahesh Kakde, and Kevin Ventullo in 2018. A further refinement of the p-adic conjecture was proposed by Gross in 1988. In 1984, John Tate formulated the Brumer–Stark conjecture, which gives a refinement of the abelian rank-one Stark conjecture at totally split finite primes (for totally complex extensions of totally real base fields). The function field analogue of the Brumer–Stark conjecture was proved by John Tate and Pierre Deligne in 1984. In 2023, Dasgupta and Kakde proved the Brumer–Stark conjecture away from the prime 2. In 1996, Karl Rubin proposed an integral refinement of the Stark conjecture in the abelian case. In 1999, Cristian Dumitru Popescu proposed a function field analogue of Rubin's conjecture and proved it in some cases.
Notes
References Burns, David; Sands, Jonathan; Solomon, David, eds. (2004), Stark's conjectures: recent work and new directions, Contemporary Mathematics, vol. 358, Providence, RI: American Mathematical Society, doi:10.1090/conm/358, ISBN 978-0-8218-3480-0, MR 2090725, archived from the original on 2012-04-26 Manin, Yuri Ivanovich (2004), "Real multiplication and noncommutative geometry (ein Alterstraum)", in Piene, Ragni; Laudal, Olav Arnfinn (eds.), The legacy of Niels Henrik Abel, Berlin, New York: Springer-Verlag, pp. 685–727, arXiv:math/0202109, Bibcode:2002math......2109M, ISBN 978-3-540-43826-7, MR 2077591 Popescu, Cristian D. (1999), "On a refined Stark conjecture for function fields", Compositio Mathematica, 116 (3): 321–367, doi:10.1023/A:1000833610462, ISSN 0010-437X, MR 1691163 Rubin, Karl (1996), "A Stark conjecture over Z for abelian L-functions with multiple zeros", Annales de l'Institut Fourier, 46 (1): 33–62, doi:10.5802/aif.1505, ISSN 0373-0956, MR 1385509 Stark, Harold M. (1971), "Values of L-functions at s = 1. I. L-functions for quadratic forms.", Advances in Mathematics, 7 (3): 301–343, doi:10.1016/S0001-8708(71)80009-9, ISSN 0001-8708, MR 0289429 Stark, Harold M. (1975), "L-functions at s = 1. II. Artin L-functions with rational characters", Advances in Mathematics, 17 (1): 60–92, doi:10.1016/0001-8708(75)90087-0, ISSN 0001-8708, MR 0382194 Stark, H. M. (1977), "Class fields and modular forms of weight one", in Serre, Jean-Pierre; Zagier, D. B. (eds.), Modular Functions of One Variable V: Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik, July 1976, Lecture Notes in Math, vol. 601, Berlin, New York: Springer-Verlag, pp. 277–287, doi:10.1007/BFb0063951, ISBN 978-3-540-08348-1, MR 0450243 Stark, Harold M. (1976), "L-functions at s = 1. III. Totally real fields and Hilbert's twelfth problem", Advances in Mathematics, 22 (1): 64–84, doi:10.1016/0001-8708(76)90138-9, ISSN 0001-8708, MR 0437501 Stark, Harold M. (1980), "L-functions at s = 1. IV. First derivatives at s = 0", Advances in Mathematics, 35 (3): 197–235, doi:10.1016/0001-8708(80)90049-3, ISSN 0001-8708, MR 0563924 Tate, John (1984), "Les conjectures de Stark sur les fonctions L d'Artin en s=0", Mathematical Programming, Progress in Mathematics, vol. 47, Boston, MA: Birkhäuser Boston, pp. 143–153, doi:10.1007/BF01580857, ISBN 978-0-8176-3188-8, MR 0782485, S2CID 13291194
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