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Journal of the Ramanujan Mathematical Society

Volume 37, Issue 3, September 2022  pp. 231–239.

Darmon points on elliptic curves over totally real fields

Authors:  Amod Agashe and Mak Trifkovic
Author institution:Department of Mathematics, Florida State University, 208 Love Building, 1017 Academic Way, Tallahassee, Florida 32306, USA

Summary:  Let F be a totally real number field with ring of integers denoted O{F} and let E be an elliptic curve over F of conductor an ideal N of O{F}. Let K be a quadratic extension of F that is not a CM field and let O be an O{F}-order of K such that Disc(O/O{F}) is coprime to N. We show that if the sign of the functional equation of E over K is −1, the discriminant of K is coprime to N, and the part of N divisible by primes that are inert in K is square-free, then one can apply either a construction of Gärtner or a construction of Greenberg to conjecturally associate to O a point on E that we call a Darmon point. This point is initially defined over a transcendental extension of K, but is conjectured to be algebraic and defined over the narrow ring class field extension of K associated to the order O; it comes with an action of the narrow class group of O, and we state a conjectural Shimura reciprocity law for this action.

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