An Analogue of Circular Units for Products of Elliptic Curves

Abstract

We construct certain elements in the motivic cohomology group, where E and E are elliptic curves over Q. When E is not isogenous to E these elements are analogous to circular units in real quadratic fields, as they come from modular parametrizations of the elliptic curves. We then find an analogue of the class-number formula for real quadratic fields, which specializes to the usual quadratic class-number formula when E and E are quadratic twists.

K1of products of Drinfeld modular curves and special values ofL-functions

“…The assumption (I 1 , I 2 ) = 1 in Theorem 5.7 is not that essential. Along the lines of the arguments in [BS04], we can prove a similar result under the weaker assumption that I 1 and I 2 have some common factors, but are not identical.…”

“…We show an analogous formula in the Drinfeld modular case with the Archimedean place being replaced by the prime ∞. More precisely, since our L-functions essentially take rational values, we have an exact formula for the value analogous to the main theorem of [BS04].…”

Regulators for Rankin–Selberg products of modular forms

“…In the weight 2 case, an explicit version of Beilinson's formula for L(f ⊗ g, 2), similar to Theorem 6.4, was proved by Baba and Sreekantan [1] and by Bertolini, Darmon and Rotger [4]. In the higher weight case, a similar formula for the regulator of generalized Beilinson-Flach elements was proved by Scholl (unpublished) and recently by Kings, Loeffler and Zerbes [17].…”