# 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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