## Numerical verification of the Stark-Chinburg conjecture for some icosahedral representations.(English)Zbl 1082.11082

The authors give a numerical verification of a variant of Stark’s conjecture which they call Stark-Chinburg conjecture. They are the first to show the existence of a certain unit (“Stark unit”) in non-abelian Galois extensions $${\mathcal K}$$ which allow two-dimensional representations of icosahedral isomorphism type in $$\text{PGL}_2(\mathbb C)$$. In this case the degree $$[{\mathcal K}:\mathbb Q]$$ is 240. The authors were successful in verifying the conjecture for 14 icosahedral representations with odd quadratic determinant. An illustrative example demonstrating the complexity of the calculations is included. For instance, they present the minimal polynomial of the fourth root of a Stark unit, a polynomial of degree 120 with large coefficients.

### MSC:

 11Y40 Algebraic number theory computations 11R42 Zeta functions and $$L$$-functions of number fields

### Keywords:

Stark conjectures; icosahedral Galois representations
Full Text:

### References:

 [1] Basmaji J., ”A Table of As-Fields with Discriminant up to 40272.” (2002) [2] Basmaji J., On Artin’s Conjecture for Odd 2-Dimensional Representations pp 37– (1994) [3] Brown K., Cohomology of Groups. (1994) [4] Buhler J., Icosahedral Galois Representations. (1978) · Zbl 0374.12002 [5] Burns D., Compositio Math. 129 pp 203– (2001) · Zbl 1014.11070 [6] Buzzard K., Duke Math Journal 109 (2) pp 283– (2001) · Zbl 1015.11021 [7] Chinburg T., Adv. in Math. 48 pp 82– (1983) · Zbl 0528.12012 [8] Cohen H., A Course in Computational Algebraic Number Theory. (1993) · Zbl 0786.11071 [9] Cohen H., Advanced Topics in Computational Number Theory. (2000) · Zbl 0977.11056 [10] Crespo T., C. R. Acad. Sci. Paris 315 pp 625– (1992) [11] Dummit D., Math. Comp. 66 pp 1239– (1997) · Zbl 0904.11033 [12] Fogel K., PhD diss., in: ”Stark’s Conjecture for Octahedral Extensions.” (1998) [13] Frey G., On Artin’s Conjecture for Odd 2-Dimensional Representations (1994) · Zbl 0801.00004 [14] Jehanne A., J. Number Theory 89 pp 340– (2001) · Zbl 0984.12003 [15] Jehanne A., J. Théorie des Nombres de Bordeaux 12 (2) pp 475– (2000) · Zbl 1161.11348 [16] Jehanne A., ”Modularity of Some Odd Icosahedral Representations.” (2001) [17] Herbrand J., C. R. Acad. Sci. Paris 191 pp 1282– (1930) [18] Herbrand J., C. R. Acad. Sci. Paris 192 pp 24– (1931) [19] Kiming I., On Artin’s Conjecture for Odd 2-Dimensional Representations pp 1– (1994) · Zbl 0839.11056 [20] Kiming I., On Artin’s Conjecture for Odd 2-Dimensional Representations pp 109– (1994) · Zbl 0839.11056 [21] Minkowski H., Gbttinger Nachrichten pp 90– (1900) [22] Batut C., ”The Number Theory System PARI.” (2003) [23] Popescu C., J. reine angew. Math. (2003) [24] Roblot X.-F., Experimental Math. 9 pp 251– (2000) · Zbl 0986.11074 [25] Rubin K., Annales de ITnstitut Fourier 46 pp 33– (1996) · Zbl 0834.11044 [26] Sands J., Adv. in Math. 66 pp 62– (1987) · Zbl 0631.12006 [27] Stark H. M., Adv. in Math. 17 pp 60– (1975) · Zbl 0316.12007 [28] Stark H. M., Adv. in Math. 22 pp 64– (1976) · Zbl 0348.12017 [29] Stark H. M., Algebraic Number Fields pp 55– (1977) [30] Stark H. M., Adv. in Math. 35 pp 197– (1980) · Zbl 0475.12018 [31] Stark H. M., Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979) pp 261– (1981) [32] Tate J. T., Les conjectures de Stark sur les fonctions L d’Artin en s = 0. (1984) · Zbl 0545.12009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.