On the characteristic polynomial of the Gross regulator matrix

Dasgupta, Samit; Spieß, Michael

doi:10.1090/tran/7393

On the characteristic polynomial of the Gross regulator matrix
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by Samit Dasgupta and Michael Spieß PDF

Trans. Amer. Math. Soc. 372 (2019), 803-827 Request permission

Abstract:

We present a conjectural formula for the principal minors and the characteristic polynomial of Gross’s regulator matrix associated to a totally odd character of a totally real field. The formula is given in terms of the Eisenstein cocycle, which was defined and studied earlier by the authors and collaborators. For the determinant of the regulator matrix, our conjecture follows from recent work of Kakde, Ventullo, and the first author. For the diagonal entries, our conjecture overlaps with the conjectural formula presented in our prior work. The intermediate cases are new and provide a refinement of the Gross–Stark conjecture.

References

M. F. Atiyah and C. T. C. Wall, Cohomology of groups, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 94–115. MR 0219512
Pierrette Cassou-Noguès, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $p$-adiques, Invent. Math. 51 (1979), no. 1, 29–59 (French). MR 524276, DOI 10.1007/BF01389911
Pierre Charollois and Samit Dasgupta, Integral Eisenstein cocycles on $\textbf {GL}_n$, I: Sczech’s cocycle and $p$-adic $L$-functions of totally real fields, Camb. J. Math. 2 (2014), no. 1, 49–90. MR 3272012, DOI 10.4310/CJM.2014.v2.n1.a2
Pierre Charollois, Samit Dasgupta, and Matthew Greenberg, Integral Eisenstein cocycles on $\mathbf {GL}_n$, II: Shintani’s method, Comment. Math. Helv. 90 (2015), no. 2, 435–477. MR 3351752, DOI 10.4171/CMH/360
Samit Dasgupta, Shintani zeta functions and Gross-Stark units for totally real fields, Duke Math. J. 143 (2008), no. 2, 225–279. MR 2420508, DOI 10.1215/00127094-2008-019
Samit Dasgupta, Henri Darmon, and Robert Pollack, Hilbert modular forms and the Gross-Stark conjecture, Ann. of Math. (2) 174 (2011), no. 1, 439–484. MR 2811604, DOI 10.4007/annals.2011.174.1.12
Samit Dasgupta, Mahesh Kakde, and Kevin Ventullo, On the Gross-Stark conjecture, Ann. of Math. (2) 188 (2018), no. 3, 833–870. MR 3866887, DOI 10.4007/annals.2018.188.3.3
Samit Dasgupta, Gross-Stark units, Stark-Heegner points, and class fields of real quadratic fields, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–University of California, Berkeley. MR 2706449
Samit Dasgupta and Michael Spieß, The Eisenstein cocycle and Gross’s tower of fields conjecture, Ann. Math. Qué. 40 (2016), no. 2, 355–376 (English, with English and French summaries). MR 3529186, DOI 10.1007/s40316-015-0046-2
Pierre Deligne and Kenneth A. Ribet, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), no. 3, 227–286. MR 579702, DOI 10.1007/BF01453237
Benedict H. Gross, $p$-adic $L$-series at $s=0$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 979–994 (1982). MR 656068
Michael Spieß, On special zeros of $p$-adic $L$-functions of Hilbert modular forms, Invent. Math. 196 (2014), no. 1, 69–138. MR 3179573, DOI 10.1007/s00222-013-0465-0
Michael Spiess, Shintani cocycles and the order of vanishing of $p$-adic Hecke $L$-series at $s=0$, Math. Ann. 359 (2014), no. 1-2, 239–265. MR 3201900, DOI 10.1007/s00208-013-0983-5
Kevin Ventullo, On the rank one abelian Gross-Stark conjecture, Comment. Math. Helv. 90 (2015), no. 4, 939–963. MR 3433283, DOI 10.4171/CMH/374