Faltings heights of CM elliptic curves and special Gamma values

1 Background In 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 integersOK 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 OL. Let E/L be an elliptic curve over L, and let E/OL be a Néron model for E/L. Let E/OL be the sheaf of Néron differentials, and let s∗ E/OL be the pullback by the zero section s : Spec(OL) → E . Choose a differential ω ∈ H0(E/L, E/L). Then the Faltings height of E/L is defined by

Remark 1.2. The Faltings height of E/L depends on the number eld L. This dependence can be eliminated by passing to a nite extension L /L such that E/L has everywhere semistable reduction. In particular, one denes the stable Faltings height by The stable Faltings height is independent of both the number eld L and the choice of extension L /L. Now, assume that E/L is a CM elliptic curve with everywhere good reduction, satisfying the hypotheses of Theorem 1.1. Moreover, assume that E has complex multiplication by the maximal order O K in K. Then the everywhere good reduction assumption implies that h stab Fal (E/L) = h Fal (E/L) and that ∆ E/L = O L . Also, since O K has conductor f = 1, then Theorem 1.1 gives ( (1.2) Deligne then observed (see [Del85,p. 29]) that the classical Chowla-Selberg formula [CS67] can be used to prove that .
(  [Kan90]. Here we give a detailed analytic proof of a Chowla-Selberg formula for orders in K. This proof is based on a renormalized Kronecker limit formula for the non-holomorphic SL 2 (Z) Eisenstein series, a period formula which relates the zeta function of an order in K to values of the Eisenstein series at CM points corresponding to classes in the ideal class group of the order, and a factorization of the zeta function of an order given by Zagier [Zag77], and in an equivalent but dierent form by Kaneko [Kan90].
2. An example of Theorem 1.1 In this section, we use Theorem 1.1 and SageMath [S + 09] to explicitly calculate both the unstable and the stable Faltings height of a CM elliptic curve dened over L = Q( be the ring of integers and let be the order of conductor f = 3 in K. Kida [Kid01] computed tables of elliptic curves with everywhere good reduction over quadratic elds. In particular, the rst entry in [Kid01, which is dened over L, has j-invariant j(E) = j, and complex multiplication by the non-maximal The norm of ∆ E/L is given by N L/Q (∆ E/L ) = 2 18 · 5 12 · 7 12 · 23 6 · 29 6 · 47 6 · 53 6 · 71 6 .
On the other hand, letting L = L( which is dened over L and isomorphic to E/L over L , has j-invariant j(E u ) = j(E) = j, and complex multiplication by the order O 3 . Moreover, E u /L has minimal discriminant ideal ∆ E u /L = O L , and thus E u /L has everywhere good reduction over L . Now, since the discriminant of K is D = −8 and the conductor of the order O 3 is f = 3, we see that ∆ 3 = −72, w −8 = 2 and h(−8) = 1. The Kronecker symbol values are χ −8 (k) = 1 for k = 1, 3 and χ −8 (k) = −1 for k = 5, 7, and hence e(3) = 0. Therefore, noting that the coecients of E/L are contained in Q(j(E)) = L, Theorem 1.1 gives us After expanding, we get h Fal (E/L) = log   2 3/4 5 1/2 7 1/2 23 1/4 29 1/4 47 1/4 53 1/4 71 1/4 6 Similarly, noting that the coecients of E u /L are also contained in Q(j(E u )) = L ⊂ L , Theorem 1.1 gives the following formula for the stable Faltings height, After expanding and using the fact that N L /Q (∆ E u /L ) = 1, we get Numerically, these values of the Faltings height are h Fal (E/L) ≈ 6.22291129399367 and h stab Fal (E/L) ≈ −0.721100481725771.

Taylor expansion of the non-Holomorphic Eisenstein series
Let H denote the complex upper half-plane and dene the stabilizer of the cusp ∞ by Then the non-holomorphic SL 2 (Z) Eisenstein series is dened by where Γ(s) is Euler's Gamma function, ζ(s) is the Riemann zeta function, σ k (n) := |n k is the k-divisor function, and K ν is the K-Bessel function of order ν. The Fourier expansion shows that E(z, s) extends to a meromorphic function on C with a simple pole at s = 1. We next make the shift s → (s + 1)/2 in the Fourier expansion of E(z, s) and calculate the Taylor expansion of the shifted Eisenstein series E(z, Then the Taylor expansions of A, B and C at s = −1 are given as follows, By combining these Taylor expansions, we get Next, recall that the Dedekind eta function is the weight 1/2 modular form for SL 2 (Z) dened by the innite product (1 − q n ), q := e 2πiz , z ∈ H.
One has the following identity relating the second term in the Taylor expansion of E(z, (s + 1)/2) at s = −1 to η(z).

Zeta functions of orders and CM values of Eisenstein series
We begin by recalling some facts regarding orders in imaginary quadratic elds (see e.g. Cox [Cox13,7]). Let K be an imaginary quadratic eld of discriminant D. Given Moreover, by [Cox13, equation (7.6)] we have a −1 = 1 a a , and thus Proposition 4.1. With notation as above, we have

s).
We will need the following lemma. αO f = βuO f = βO f , and hence αa = βa. This proves that φ is well-dened.
To prove that φ is surjective, suppose that I ∈ [a] with I ⊂ O f . Then I = αa for some α ∈ K × , or equivalently, We now prove Proposition 4.1.

A Chowla-Selberg formula for imaginary quadratic orders
In this section we will prove the following theorem.
Theorem 5.1. With notation as in Section 4, we have .
Before proving Theorem 5.1, we demonstrate how it can be used to explicitly evaluate a CM value of η(z). We then numerically verify the resulting identity using SageMath [S + 09].
Using SageMath, one can check that both sides of the previous equality are approximately 0.592382781332416, which serves as a numerical verication of the identity in Theorem 5.1.
Proof of Theorem 5.1: By Proposition 4.1, we have Then summing over all ideal classes in Cl(O f ) yields For convenience, dene the function E (z a −1 , (s + 1)/2) .
It remains to prove the following lemma.
Lemma 5.3. We have
Then L f (s) can be written as Now, recall that ∆ E/L is the minimal discriminant of E/L and j(E) is the j-invariant of E/L.
where F (z) is dened in equation (3.4) and z a −1 is a CM point as in equation (4.1).