Abstract
This is the second in a pair of papers about residually reducible Galois deformation rings with non-optimal level. In the first paper, we proved a Galois-theoretic criterion for the deformation ring to be as small as possible. This paper focuses on the computations needed to verify this criterion. We adapt a technique developed by Sharifi to compute number fields with twisted-Heisenberg Galois group and prescribed ramification, and compute the splitting behavior of primes in these extensions.
Similar content being viewed by others
Data availability
All data generated or analysed during this study are included in this published article (and its supplementary information files).
Notes
Note that \(b_i|_{I_{\ell _i}}: I_{\ell _i} \rightarrow {\mathbb {F}}_p(1)\) is a well-defined homomorphism because \(I_{\ell _i}\) acts trivially on \({\mathbb {F}}_p(1))\).
The Strelka Computer Cluster is located at Swarthmore College. Its technical specifications can be found at https://kb.swarthmore.edu/display/ACADTECH/Strelka+Computer+Cluster.
The HTC Cluster is located at the Center for Research Computing at the University of Pittsburgh. Its technical specifications can be found at https://crc.pitt.edu/resources.
This problem is in the process of being fixed by Sage and Pari developers and is logged at https://trac.sagemath.org/ticket/31327.
References
Calegari, F., Emerton, M.: On the ramification of Hecke algebras at Eisenstein primes. Invent. Math. 160(1), 97–144 (2005)
Fontaine, J.-M.: Il n’y a pas de variété abélienne sur Z. Invent. Math. 81(3), 515–538 (1985)
Hsu, C., Wake, P., Wang-Erickson, C.: Explicit non-Gorenstein \(R={\mathbb{T}}\) via rank bounds I: Deformation theory (2022). Preprint, arXiv: 2209.00536 [math.NT]
Kraines, D.: Massey higher products. Trans. Am. Math. Soc. 124, 431–449 (1966)
Lecouturier, E.: On the Galois structure of the class group of certain Kummer extensions. J. Lond. Math. Soc. (2) 98(1), 35–58 (2018)
Lemmermeyer, F.: Reciprocity laws. Springer Monographs in Mathematics. From Euler to Eisenstein. Springer, Berlin (2000)
Lam, Y.H.J., Liu, Y., Sharifi, R., Wake, P., Wang, J.: Generalized Bockstein maps and Massey products (2021). arXiv:2004.11510v2 [math.NT]
May, J.P.: Matric Massey products. J. Algebra 12, 533–568 (1969)
Merel, L.: L’accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de \(J_0(p)\). J. Reine Angew. Math. 477, 71–115 (1996)
Neukirch, J.: Algebraic Number Theory, volume 322 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin: Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G, Harder (1999)
Stein, W.A., et al.: SageMath, the Sage Mathematics Software System (accessed online through CoCalc). The Sage Development Team (2018). http://www.sagemath.org, https://cocalc.com
Serre, J.-P.: Local Fields, vol. 67 of Graduate Texts in Mathematics. Springer, New York. Translated from the French by Marvin Jay Greenberg (1979)
Serre, J.-P.: Sur les représentations modulaires de degré \(2\) de \({\rm Gal}(\overline{\textbf{ Q} }/{\textbf{ Q} })\). Duke Math. J. 54(1), 179–230 (1987)
Sharifi, R.T.: Twisted Heisenberg representations and local conductors. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), The University of Chicago (1999)
Sharifi, R.T.: Massey products and ideal class groups. J. Reine Angew. Math. 603, 1–33 (2007)
Tahara, K.: On the second cohomology groups of semidirect products. Math. Z. 129, 365–379 (1972)
The PARI Group, Univ. Bordeaux. PARI/GP version 2.13.4 (2022). http://pari.math.u-bordeaux.fr/
Wake, P.: The Eisenstein ideal for weight \(k\) and a Bloch-Kato conjecture for tame families. J. Eur. Math. Soc. (2022). https://doi.org/10.4171/JEMS/1251
Wake, P., Wang-Erickson, C.: The rank of Mazur’s Eisenstein ideal. Duke Math. J. 169(1), 31–115 (2020)
Acknowledgements
The first-named author thanks the University of Bristol and the Heilbronn Institute for Mathematical Research for its partial support of this project. The second-named author was supported in part by NSF grant DMS-1901867, and would like to thank his coauthors on the paper [7]; that paper inspired many of the ideas used here about how to compute Massey products. The third-named author was supported in part by Simons Foundation award 846912, and thanks the Department of Mathematics of Imperial College London for its partial support of this project from its Mathematics Platform Grant. We also thank John Cremona for several insightful conversations about computational aspects of this project as well as the referee for their helpful comments and suggestions. This research was supported in part by the University of Pittsburgh Center for Research Computing and Swarthmore College through the computing resources provided.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Additional file
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hsu, C., Wake, P. & Wang-Erickson, C. Explicit non-Gorenstein \(R={\mathbb {T}}\) via rank bounds II: Computational aspects. Res. number theory 9, 16 (2023). https://doi.org/10.1007/s40993-022-00401-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-022-00401-1