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Explicit non-Gorenstein \(R={\mathbb {T}}\) via rank bounds II: Computational aspects

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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.

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All data generated or analysed during this study are included in this published article (and its supplementary information files).

Notes

  1. 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 \({\mathbb {F}}_p(1))\).

  2. 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.

  3. 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.

  4. 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

  1. Calegari, F., Emerton, M.: On the ramification of Hecke algebras at Eisenstein primes. Invent. Math. 160(1), 97–144 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fontaine, J.-M.: Il n’y a pas de variété abélienne sur Z. Invent. Math. 81(3), 515–538 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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]

  4. Kraines, D.: Massey higher products. Trans. Am. Math. Soc. 124, 431–449 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lecouturier, E.: On the Galois structure of the class group of certain Kummer extensions. J. Lond. Math. Soc. (2) 98(1), 35–58 (2018)

  6. Lemmermeyer, F.: Reciprocity laws. Springer Monographs in Mathematics. From Euler to Eisenstein. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  7. Lam, Y.H.J., Liu, Y., Sharifi, R., Wake, P., Wang, J.: Generalized Bockstein maps and Massey products (2021). arXiv:2004.11510v2 [math.NT]

  8. May, J.P.: Matric Massey products. J. Algebra 12, 533–568 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    MathSciNet  MATH  Google Scholar 

  10. 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)

  11. 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

  12. Serre, J.-P.: Local Fields, vol. 67 of Graduate Texts in Mathematics. Springer, New York. Translated from the French by Marvin Jay Greenberg (1979)

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Sharifi, R.T.: Twisted Heisenberg representations and local conductors. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), The University of Chicago (1999)

  15. Sharifi, R.T.: Massey products and ideal class groups. J. Reine Angew. Math. 603, 1–33 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tahara, K.: On the second cohomology groups of semidirect products. Math. Z. 129, 365–379 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. The PARI Group, Univ. Bordeaux. PARI/GP version 2.13.4 (2022). http://pari.math.u-bordeaux.fr/

  18. 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

    Article  Google Scholar 

  19. Wake, P., Wang-Erickson, C.: The rank of Mazur’s Eisenstein ideal. Duke Math. J. 169(1), 31–115 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Swarthmore College, Swarthmore, PA, 19081, USA

    Catherine Hsu

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Preston Wake

  3. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Carl Wang-Erickson

Authors
  1. Catherine Hsu
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  2. Preston Wake
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  3. Carl Wang-Erickson
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Correspondence to Catherine Hsu.

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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

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