No CrossRef data available.
Article contents
On the root of unity ambiguity in a formula for the Brumer–Stark units
Published online by Cambridge University Press: 27 December 2023
Abstract
We prove a conjectural formula for the Brumer–Stark units. Dasgupta and Kakde have shown the formula is correct up to a bounded root of unity. In this paper, we resolve the ambiguity in their result. We also remove an assumption from Dasgupta–Kakde’s result on the formula.
MSC classification
- Type
- Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
References
Bullach, D., Burns, D., Daoud, A., and Seo, S., Dirichlet
$L$
-series at
$s=0$
and the scarcity of Euler systems. Preprint, 2021. arXiv:2111.14689
Google Scholar
Burns, D.,
Congruences between derivatives of abelian
$L$
-functions at
$s=0$
. Invent. Math. 169(2007), no. 3, 451–499.CrossRefGoogle Scholar
Cassou-Noguès, P.,
Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta
$p$
-adiques
. Invent. Math. 51(1979), no. 1, 29–59.CrossRefGoogle Scholar
Dasgupta, S.,
Shintani zeta functions and Gross–Stark units for totally real fields
. Duke Math. J. 143(2008), no. 2, 225–279.CrossRefGoogle Scholar
Dasgupta, S. and Honnor, M. H., On the equality of three formulas for the Brumer–Stark units. Preprint, 2022. arXiv:2211.01715
Google Scholar
Dasgupta, S. and Kakde, M., Brumer–Stark units and explicit class field theory. Duke Math. J. Preprint, 2021. arXiv:2103.02516
Google Scholar
Dasgupta, S. and Kakde, M.,
On the Brumer–Stark conjecture
. Ann. of Math. (2) 197(2023), no. 1, 289–388.CrossRefGoogle Scholar
Dasgupta, S., Kakde, M., and Silliman, J., On the equivariant Tamagawa number conjecture. Preprint, 2023. arXiv:2312.09849
Google Scholar
Dasgupta, S., Kakde, M., Silliman, J., and Wang, J., The Brumer–Stark conjecture over
$\mathbb{Z}$
. Preprint, 2023. arXiv:2310.16399
CrossRefGoogle Scholar
Dasgupta, S. and Spieß, M.,
Partial zeta values, Gross’s tower of fields conjecture, and Gross–Stark units
. J. Eur. Math. Soc. (JEMS) 20(2018), no. 11, 2643–2683.CrossRefGoogle Scholar
Dasgupta, S. and Spiess, M.,
On the characteristic polynomial of the Gross regulator matrix
. Trans. Amer. Math. Soc. 372(2019), no. 2, 803–827.CrossRefGoogle Scholar
Deligne, P. and Ribet, K. A.,
Values of abelian
$L$
-functions at negative integers over totally real fields
. Invent. Math. 59(1980), no. 3, 227–286.CrossRefGoogle Scholar
Gross, B. H.,
$p$
-adic
$L$
-series at
$s=0$
. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(1982), no. 3, 979–994.Google Scholar
Gross, B. H.,
On the values of abelian
$L$
-functions at
$s=0$
. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35(1988), no. 1, 177–197.Google Scholar
Tate, J.,
On Stark’s conjectures on the behavior of
$L(s,\chi)$
at
$s=0$
. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(1982), no. 3, 963–978.Google Scholar