# Faltings heights of CM elliptic curves and special Gamma values

- Adrian Barquero-Sanchez
^{1}, - Lindsay Cadwallader
^{2}, - Olivia Cannon
^{3}, - Tyler Genao
^{4}and - Riad Masri
^{5}Email author

**Received: **25 August 2016

**Accepted: **13 February 2017

**Published: **5 June 2017

## 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.

## 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 integers \(\mathcal {O}_K\) of an imaginary quadratic field *K* in terms of Euler’s Gamma function \(\Gamma (s)\) 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 \(\mathbb {Q}\) with complex multiplication by a non-maximal order (see Sect. 2).

*L*be a number field with ring of integers \(\mathcal {O}_L\). Let

*E*/

*L*be an elliptic curve over

*L*, and let \(\mathcal {E}/\mathcal {O}_L\) be a Néron model for

*E*/

*L*. Let \(\Omega _{\mathcal {E}/\mathcal {O}_L}\) be the sheaf of Néron differentials, and let \(s^{*}\Omega _{\mathcal {E}/\mathcal {O}_L}\) be the pullback by the zero section \(s : \text {Spec}(\mathcal {O}_L) \rightarrow \mathcal {E}\). Choose a differential \(\omega \in H^0(E/L, \Omega _{E/L})\). Then the

*Faltings height*of

*E*/

*L*is defined by

To state our main results, we fix the following notation. Let *K* be an imaginary quadratic field of discriminant *D* with ideal class group \(\text {Cl}(D)\), unit group \(\mathcal {O}_K^{\times }\), and Kronecker symbol \(\chi _D\). Let \(h(D)=\#\text {Cl}(D)\) be the class number and \(w_D=\# \mathcal {O}_K^{\times }\) be the number of units. For an elliptic curve *E* / *L*, let \(\Delta _{E/L}\) be the minimal discriminant ideal and *j*(*E*) be the *j*-invariant.

### Theorem 1.1

*E*/

*L*be an elliptic curve with complex multiplication by an order \(\mathcal {O}_f \subset K\) of conductor \(f \in \mathbb {Z}^{+}\) and discriminant \(\Delta _f=f^2D\). Assume that the coefficients of the Weierstrass equation for

*E*/

*L*are contained in \(\mathbb {Q}(j(E))\). Then the Faltings height of

*E*/

*L*is given by

### Remark 1.2

Our assumption that the coefficients of the Weierstrass equation for *E* / *L* be contained in \(\mathbb {Q}(j(E))\) is used crucially in the proof of Proposition 6.1, which is an important component in the proof of Theorem 1.1. This hypothesis can be removed if we instead work with the *stable* Faltings height, which we do in Theorem 1.3.

*L*, then it is not necessarily true that \(h_{\text {Fal}}(E/L)=h_{\text {Fal}}(E/L^{\prime })\). However, if an elliptic curve over a number field has everywhere semistable reduction, then the Faltings height is invariant under finite field extensions. This leads one to define the

*stable Faltings height*of

*E*/

*L*by

*L*such that \(E/L^{\prime }\) has everywhere semistable reduction.

### Theorem 1.3

*E*/

*L*be an elliptic curve with complex multiplication by an order \(\mathcal {O}_f \subset K\) of conductor \(f \in \mathbb {Z}^{+}\) and discriminant \(\Delta _f=f^2D\). Then the stable Faltings height of

*E*/

*L*is given by

### Remark 1.4

*E*/

*L*has complex multiplication by the maximal order \(\mathcal {O}_K\) in

*K*. Since \(\mathcal {O}_K\) has conductor \(f = 1\), by Theorem 1.3 we have

*geometric height*of

*E*and denoted by \(h_{\text {geom}}(E)\). It can be shown that

An important component in the proof of Theorem 1.1 is a Chowla–Selberg formula for any order in *K*. An arithmetic-geometric proof of such a formula was given by Nakkajima and Taguchi [13] by employing a theorem of Faltings which relates the Faltings heights of two isogenous abelian varieties. Kaneko briefly outlined an analytic approach to the same formula in the research announcement [7]. 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 Eisenstein series on \(SL_2(\mathbb {Z})\), 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 [19], and in an equivalent but different form by Kaneko [7].

## 2 Examples

In this section, we use Theorems 1.1 and 1.3, and SageMath [14] to evaluate the unstable and stable Faltings height of an elliptic curve over \(\mathbb {Q}\) with complex multiplication by a non-maximal order.

### Example 2.1

*K*. Let \(A = 255\) and consider the elliptic curve (see [8, Eq. (2.2)])

*j*-invariant of \(E_A/\mathbb {Q}\) is \(j = A^3 = 255^3\).

We now use Theorems 1.1 and 1.3 to evaluate the unstable and stable Faltings height of \(E_A/\mathbb {Q}\).

*K*is \(D = -7\) and the conductor of the order \(\mathcal {O}_2\) is \(f = 2\), we have \(\Delta _2 = -28\). Also, \(w_{-7} = 2\) and \(h(-7) = 1\). The Kronecker symbol values are \(\chi _{-7}(k) = 1\) for \(k = 1, 2, 4\) and \(\chi _{-7}(k) = -1\) for \(k = 3, 5, 6\). The only prime

*p*|

*f*is \(p = 2\), and we have

*L*(given by the same Weierstrass equation). Then the quartic twist of \(E_A/\mathbb {Q}\) by

*u*is the elliptic curve

^{1}

*not*contained in \(\mathbb {Q}(j) = \mathbb {Q}\), we cannot apply Theorem 1.1 directly to evaluate the Faltings height of \(E_A^u/L(\sqrt{u})\). This demonstrates the usefulness of Theorem 1.3 in evaluating the stable Faltings height of a CM elliptic curve.

## 3 The Kronecker limit formula

*k*-divisor function, and \(K_\nu \) is the

*K*-Bessel function of order \(\nu \). The Fourier expansion shows that

*E*(

*z*,

*s*) extends to a meromorphic function on \(\mathbb {C}\) with a simple pole at \(s=1\).

*E*(

*z*,

*s*) and calculate the Taylor expansion of the shifted Eisenstein series \(E(z,(s+1)/2)\) at \(s=-1\) to get

*Dedekind eta function*is the weight 1 / 2 modular form for \(SL_2(\mathbb {Z})\) defined by the infinite product

*F*(

*z*) is the \(SL_2(\mathbb {Z})\)-invariant function defined by

## 4 Zeta functions of orders and CM values of Eisenstein series

In this section, we relate the zeta function of an order in an imaginary quadratic field to values of the Eisenstein series *E*(*z*, *s*) at CM points corresponding to classes in the ideal class group of the order.

*K*be an imaginary quadratic field of discriminant

*D*. Given \(f \in \mathbb {Z}^{+}\), let \(\mathcal {O}_f\) be the (unique) order of conductor

*f*in

*K*. A fractional \(\mathcal {O}_f\)-ideal \(\mathfrak {a}\) is a subset of

*K*which is a non-zero finitely generated \(\mathcal {O}_f\)-module. A fractional \(\mathcal {O}_f\)-ideal \(\mathfrak {a}\) is

*proper*if

*ideal class group*of \(\mathcal {O}_f\) is defined as the quotient group

*Dedekind zeta function*of \(\mathcal {O}_f\) is defined by

*ideal class zeta function*by

*K*. Then

Let \(\mathcal {O}^{\times }_f\) be the group of units in \(\mathcal {O}_f\), and let \(w_f=\# \mathcal {O}^{\times }_f\).

### Proposition 4.1

We will need the following lemma.

### Lemma 4.2

### Proof

We first prove that the map \(\phi \) is well-defined. Observe that if \(\alpha \in \mathfrak {a}^{-1}\), then \(\alpha \mathfrak {a} \subseteq \mathcal {O}_f\) since \(\mathfrak {a}^{-1}\mathfrak {a}=\mathcal {O}_f\). Next, observe that if \([\alpha ] = [\beta ]\), then \(\alpha =\beta u\) for some unit \(u \in \mathcal {O}_f^{\times }\). It follows that \(\alpha \mathcal {O}_f =\beta u\mathcal {O}_f=\beta \mathcal {O}_f\), and hence \(\alpha \mathfrak {a}=\beta \mathfrak {a}\).

To prove that \(\phi \) is injective, suppose that \(\alpha \mathfrak {a}=\beta \mathfrak {a}\). Then \(\alpha \mathfrak {a} \mathfrak {a}^{-1}=\beta \mathfrak {a}\mathfrak {a}^{-1}\), which implies that \(\alpha \mathcal {O}_f =\beta \mathcal {O}_f\), or equivalently, that \([\alpha ] = [\beta ]\).

To prove that \(\phi \) is surjective, suppose that \(I \in [{\mathfrak {a}}]\) with \(I \subset \mathcal {O}_f\). Then \(I=\alpha \mathfrak {a}\) for some \(\alpha \in K^{\times }\), or equivalently, \(I\mathfrak {a}^{-1}=\alpha \mathcal {O}_f\). Since *I* is integral, we have \(I\mathfrak {a}^{-1} \subset \mathfrak {a}^{-1}\), so that \(\alpha \in \mathfrak {a}^{-1}\). Then \([\alpha ] \in (\mathfrak {a}^{-1}\setminus \{0\})/\mathcal {O}_f^{\times }\) with \(\phi ([\alpha ])=\alpha \mathfrak {a}=I\). \(\square \)

We now prove Proposition 4.1.

### Proof of Proposition 4.1

## 5 A Chowla–Selberg formula for imaginary quadratic orders

In this section, we will prove the following result.

### Theorem 5.1

*F*(

*z*) is defined by (3.2), \(z_{{\mathfrak {a}}^{-1}}\) is a CM point as in (4.1), and

Before proving Theorem 5.1, we illustrate how it can be used to evaluate the Dedekind eta function \(\eta (z)\) at CM points.

### Example 5.2

*K*given by

*K*is \(D = -4\), the discriminant of \(\mathcal {O}_2\) is \(\Delta _2=2^2(-4)=-16\). Also, \(h(-4) = 1\) and \(w_{-4}=4\). We have \(h(\mathcal {O}_2)=1\), so that \({\text {Cl}}(\mathcal {O}_2)=\lbrace [\mathcal {O}_2]\rbrace \). Then since \(\mathcal {O}_2^{-1} = \mathcal {O}_2 = \mathbb {Z}+ \mathbb {Z}2i\), from (4.1) we can take \(z_{\mathcal {O}_2^{-1}} = 2i\) for the CM point. It follows that

*p*|

*f*is \(p = 2\), and we have

### Proof of Theorem 5.1

*e*(

*p*) is defined by (5.1).

## 6 Faltings heights of CM elliptic curves

In this section, we will prove the following result which is based on Silverman [16, Proposition 1.1].

### Proposition 6.1

*E*/

*L*be an elliptic curve with complex multiplication by an order \(\mathcal {O}_f\) in an imaginary quadratic field

*K*. Assume that the coefficients of the Weierstrass equation for

*E*/

*L*are contained in \(\mathbb {Q}(j(E))\). Then

*F*(

*z*) is defined by (3.2) and \(z_{{\mathfrak {a}}^{-1}}\) is a CM point as in (4.1).

### Proof

*E*/

*L*is given by

*E*/

*L*has coefficients in \(\mathbb {Q}(j(E))\), then for each fixed \(\tau \in \text {Hom}(\mathbb {Q}(j(E)), \mathbb {C})\) we can take the same point \(z_{\sigma } \in {\mathbb H}\) in the isomorphism (6.1) for all \(\sigma \in \text {Hom}(L, \mathbb {C})\) such that \(\sigma |_{\mathbb {Q}(j(E))}=\tau \). Therefore, if we let \(\sigma _{\tau } \in \text {Hom}(L, \mathbb {C})\) denote any of the \([L:\mathbb {Q}(j(E))]\) embeddings which extend \(\tau \in \text {Hom}(\mathbb {Q}(j(E)), \mathbb {C})\), then we have

*F*(

*z*) is \(SL_2(\mathbb {Z})\)-invariant, it follows that

## 7 Proofs of Theorem 1.1 and Theorem 1.3

In this section, we prove Theorems 1.1 and 1.3.

### Proof of Theorem 1.1

### Proof of Theorem 1.3

*E*/

*L*has complex multiplication, the

*j*-invariant

*j*(

*E*) is an algebraic integer. Hence by [18, Proposition VII.5.5],

*E*/

*L*has potential good reduction. Accordingly, let \(L^{\prime }/L\) be a finite extension such that \(E/L^{\prime }\) has everywhere good reduction. Now, by [18, Proposition III.1.4], there is a finite extension \(L^{\prime \prime }/L^{\prime }\) with \(\mathbb {Q}(j(E)) \subset L^{\prime \prime }\) and an elliptic curve \(E/L^{\prime \prime }\) such that \(E/L^{\prime \prime }\) is given by a Weierstrass equation with coefficients in \(\mathbb {Q}(j(E))\) and such that \(E/L^{\prime }\) is isomorphic to \(E/L^{\prime \prime }\). By the semistable reduction theorem [18], Proposition VII.5.4 (b)], the curve \(E/L^{\prime \prime }\) also has everywhere good reduction. Therefore we have \(\Delta _{E/L^{\prime \prime }} = \mathcal {O}_{L^{\prime \prime }}\). Finally, since \(N_{L^{\prime \prime }/\mathbb {Q}}(\Delta _{E/L^{\prime \prime }}) = 1\), then Theorem 1.3 follows by applying Theorem 1.1 to \(E/L^{\prime \prime }\) and observing that

## Declarations

### Acknowledgements

The authors would like to thank Matt Papanikolas for several helpful discussions, and the referee for many suggestions which greatly improved the paper. The authors were partially supported by the NSF Grants DMS-1162535 and DMS-1460766, and the University of Costa Rica, during the preparation of this work.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Arakawa, T., Ibukiyama, T., Kaneko, M.: Bernoulli numbers and zeta functions. With an appendix by Don Zagier. In: Springer monographs in mathematics. Springer, Tokyo (2014)Google Scholar
- Chowla, S., Selberg, A.: On Epstein’s zeta-function. J. Reine Angew. Math.
**227**, 86–110 (1967)MathSciNetMATHGoogle Scholar - Cox, D.A.: Primes of the form $$x^2+ny^2$$ x 2 + n y 2 . Fermat, class field theory and complex multiplication. In: Pure and applied mathematics (Hoboken), 2nd edn. Wiley, Hoboken (2013)Google Scholar
- Deitmar A.: Automorphic forms. Translated from the 2010 German original. Universitext. Springer, London (2013)Google Scholar
- Deligne, P.: Preuve des conjectures de Tate et de Shafarevitch (d’après G. Faltings). (French) Proof of the Tate and Shafarevich conjectures (after G. Faltings). Seminar Bourbaki, vol. 1983/84, pp. 25–41. Astérisque No. 121–122 (1985)Google Scholar
- Faltings, G.: Finiteness theorems for abelian varieties over number fields. Translated from the German original [Invent. Math. 73(3), 1983, pp. 349–366; ibid. 75 (1984), no. 2, 381] by Edward Shipz. Arithmetic geometry (Storrs, Conn., 1984), 9–27. Springer, New York (1986)Google Scholar
- Kaneko, M.: A generalization of the Chowla–Selberg formula and the zeta functions of quadratic orders. Proc. Jpn. Acad. Ser. A Math. Sci.
**66**, 201–203 (1990)MathSciNetView ArticleMATHGoogle Scholar - Kida, M.: Computing elliptic curves having good reduction everywhere over quadratic fields. Tokyo J. Math.
**24**, 545–558 (2001)MathSciNetView ArticleMATHGoogle Scholar - Kurokawa, N., Kurihara, M., Saito, T.: Number theory. 3. Iwasawa theory and modular forms. Translated from the Japanese by Masato Kuwata. Translations of Mathematical Monographs, vol. 242. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence (2012)Google Scholar
- Lang, S.: Elliptic functions. With an appendix by J. Tate, 2nd edn. In: Graduate texts in mathematics, vol. 112. Springer, New York (1987)Google Scholar
- Lerch, M.: Sur quelques formules relatives au nombre des classes. Bull. Sei. Math.
**21**, 302–303 (1897)Google Scholar - Milne, J.S.: Abelian varieties (v2.00). (2008). www.jmilne.org/math/
- Nakkajima, Y., Taguchi, Y.: A generalization of the Chowla–Selberg formula. J. Reine Angew. Math.
**419**, 119–124 (1991)MathSciNetMATHGoogle Scholar - The Sage Developers. SageMath, the Sage Mathematics Software System (Version 6.10). The Sage Developers, Newcastle upon Tyne. (2015). http://www.sagemath.org
- Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Reprint of the 1971 original. Publications of the Mathematical Society of Japan, vol. 11. Kanô Memorial Lectures, 1. Princeton University Press, Princeton (1994)Google Scholar
- Silverman, J.H.: Heights and elliptic curves. Arithmetic geometry (Storrs, Conn., 1984), pp. 253–265. Springer, New York (1986)Google Scholar
- Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 151. Springer, New York (1994)Google Scholar
- Silverman, J.H.: The arithmetic of elliptic curves, 2nd ed. In: Graduate texts in mathematics, vol. 106. Springer, Dordrecht (2009)Google Scholar
- Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular functions of one variable, VI, Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976, pp. 105–169. Lecture Notes in Math., vol. 627, Springer, Berlin (1977)Google Scholar