In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

In this paper, we use methods from the spectral theory of automorphic forms to give an asymptotic formula with a power saving error term for Andrews' smallest parts function spt(n). We use this formula to deduce an asymptotic formula with a power saving error term for the number of 2-marked Durfee symbols associated to partitions of n. Our method requires that we count the number of Heegner points of discriminant −D < 0 and level N inside an "expanding" rectangle contained in a fundamental domain for Γ 0 (N).

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.

Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n. We construct a theta series θ K ∈ M d,n , where M d,n is a space of modular forms defined in terms of n and d. Moreover, if d is square free and n is at most 4 then θ K is a complete invariant for K. We also investigate whether or not the collection of θ-series, associated to the set of isomorphism classes of quartic number fields of a fixed square free discriminant d, is a linearly independent subset of M d,4 . This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.

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