Mathematische Zeitschrift

, Volume 280, Issue 3, pp 1015–1029

Quadratic \({\mathbb {Q}}\)-curves, units and Hecke \(L\)-values


DOI: 10.1007/s00209-015-1463-2

Cite this article as:
Dummigan, N. & Golyshev, V. Math. Z. (2015) 280: 1015. doi:10.1007/s00209-015-1463-2


We show that if \(K\) is a quadratic field, and if there exists a quadratic \({\mathbb {Q}}\)-curve \(E/K\) of prime degree \(N\), satisfying weak conditions, then any unit \(u\) of \(O_K\) satisfies a congruence \(u^r\equiv 1\pmod {N}\), where \(r={\mathrm {g.c.d.}}(N-1,12)\). If \(K\) is imaginary quadratic, we prove a congruence, modulo a divisor of \(N\), between an algebraic Hecke character \(\tilde{\psi }\) and, roughly speaking, the elliptic curve. We show that this divisor then occurs in a critical value \(L(\tilde{\psi },2)\), by constructing a non-zero element in a Selmer group and applying a theorem of Kato.


\({\mathbb {Q}}\)-curve Real quadratic units Hecke \(L\)-function 

Mathematics Subject Classification

11G05 11R11 11G40 

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.Algebra and Number Theory SectorInstitute for Information Transmission ProblemsMoscowRussia

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