Faltings heights of CM elliptic curves and special Gamma values

Abstract

In this paper, we evaluate the Faltings height of an elliptic curve with complex multiplication by an order in an imaginary quadratic field in terms of Euler's Gamma function at rational arguments. BackgroundIn the Seminar Bourbaki article [5], Deligne used the Chowla-Selberg formula [2] to evaluate the stable Faltings height of an elliptic curve with complex multiplication by the ring of integers O K of an imaginary quadratic field K in terms of Euler's Gamma function (s) at rational arguments. He then used this result to calculate the minimum value attained by the stable Faltings height. In this paper, we will establish a similar formula for both the unstable and stable Faltings height of an elliptic curve with complex multiplication by any order in K (not necessarily maximal). We illustrate these results by explicitly evaluating the Faltings height of an elliptic curve over Q with complex multiplication by a non-maximal order (see Sect. 2).We begin by recalling the definition of the (unstable) Faltings height of an elliptic curve, following ([12], Chapter IV, Sect. 6). Let L be a number field with ring of integers O L . Let E/L be an elliptic curve over L, and let E/O L be a Néron model for E/L.

On Kronecker terms over global function fields

“…This formula provides a very interesting arithmetic interpretation of the geometric invariant in question, took a stand near the center of arithmetic geometry ever since its discovery (cf. [5], [26], [35], [36], [1], and [3]). Here we apply Theorem 1.1 to derive an analogue of the Colmez formula for the stable "Taguchi height" of Drinfeld modules.…”