Abstract
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.
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Dummigan, N., Golyshev, V. Quadratic \({\mathbb {Q}}\)-curves, units and Hecke \(L\)-values. Math. Z. 280, 1015–1029 (2015). https://doi.org/10.1007/s00209-015-1463-2
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DOI: https://doi.org/10.1007/s00209-015-1463-2