Elliptic unit

A system of elliptic units may be constructed for an elliptic curve E with complex multiplication by the ring of integers R of an imaginary quadratic field F. For simplicity we assume that F has class number one. Let a be an ideal of R with generator α. For a Weierstrass model of E, define

${\displaystyle \Theta _{\mathbf {a} }(P)=\alpha ^{-12}\Delta _{E}^{N\mathbf {a} -1}\prod _{\mathbf {a} P=0,P\neq 0}(x-x(P))^{-6}\ .}$

where P is a point on E, Δ is the discriminant, and x is the X-coordinate on the Weierstrass model. The function Θ is independent of the choice of model, and is defined over the field of definition of E.

Let b be an ideal of R coprime to a and Q an R-generator of the b-torsion. Then Θ_a(Q) is defined over the ray class field K(b), and if b is not a prime power then Θ_a(Q) is a global unit: if b is a power of a prime p then Θ_a(Q) is a unit away from p.

The function Θ_a satisfies a distribution relation for b = (β) coprime to a:

${\displaystyle \prod _{\mathbf {b} Q=0}\Theta _{\mathbf {a} }(P+R)=\Theta _{\mathbf {a} }(\beta P)\ .}$

• Modular unit

• Coates, J.H.; Greenberg, R.; Ribet, K.A.; Rubin, K. (1999). Arithmetic Theory of Elliptic Curves. Lecture Notes in Mathematics. Vol. 1716. Springer-Verlag. ISBN 3-540-66546-3.
• Coates, John; Wiles, Andrew (1977). "On the conjecture of Birch and Swinnerton-Dyer". Inventiones Mathematicae. 39 (3): 223–251. doi:10.1007/BF01402975. Zbl 0359.14009.
• Kubert, Daniel S.; Lang, Serge (1981). Modular units. Grundlehren der Mathematischen Wissenschaften. Vol. 244. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90517-4. MR 0648603. Zbl 0492.12002.
• Robert, Gilles Unités elliptiques. (Elliptic units) Bull. Soc. Math. France, Supp. Mém. No. 36. Bull. Soc. Math. France, Tome 101. Société Mathématique de France, Paris, 1973. 77 pp.