D. Zagier discovered in 1975 [HZ] the following famous modular form of weight :

Here τ = u+iv is in the upper half plane, q = e2πiτ , H0(m) is the Hurwitz class number of binary quadratic forms of discriminant −m, and

This function can be obtained, via analytic continuation, as a special value of an Eisenstein series ε(τ, s) at s = ½. In this note, we will give an arithmetic interpretation to Zagier's Eisenstein series and its derivative at s = ½, using Arakelov theory.

Let M be the Deligne–Rapoport compactification of the moduli stack over Z of elliptic curves [DR]. In Section 3 we will define a generating function of arithmetic Chow cycles of codimension 1 in M with real coefficients, in the sense of Bost and Kühn [Bos1,Kun]:

such that

and

Here is a normalized metrized Hodge bundle onM, to be defined in Section 3, and is the Gillet–Soule intersection pairing ([GS]; see also section 2). Bost's arithmetic Chow cycles with real coefficients are crucial here since, for example, the negative Fourier coefficients of Zagier's Eisenstein series are clearly not rational numbers. This note is a slight variation of joint work with Stephen Kudla and Michael Rapoport [KRY2]. It confirms a conjecture of Kudla [Ku2].

1. The Chowla–Selberg Formula

Part of the formulas (0.4) and (0.5) can be viewed as a generalization of the Chowla–Selberg formula which we now describe.

For a positive integer m≠= 0, let and let Om be the order in Km of discriminant −m. Notice that Om exists if and only if m ≡ 0,−1 mod4. We can and will write m = dn2 such that −d is the fundamental discriminant of Km and n ≥ 1 is an integer.

Let Z(m) be the set of isomorphic classes of elliptic curves E over C such that there is an embedding Om _→ End(E). When Om does not exist, we take Z(m) to be empty. Let