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.

No abstract

We develop tools for the analysis of fronts, pulses, and wave trains in spatially extended systems with nonlocal coupling. We first determine Fredholm properties of linear operators, thereby identifying pointwise invertibility of the principal part together with invertibility at spatial infinity as necessary and sufficient conditions. We then build on the Fredholm theory to construct center manifolds for nonlocal spatial dynamics under optimal regularity assumptions, with reduced vector fields and phase space identified a posteriori through the shift on bounded solutions. As an application, we establish uniqueness of small periodic wave trains in a Lyapunov center theorem using only C 1 -regularity of the nonlinearity.