[Paper Review] Derivatives of Eisenstein series and Faltings heights

Stephen S. Kudla, Michael Rapoport+1 more|ArXiv.org|Oct 26, 2001

Advanced Algebra and Geometry19 references4 citations

TL;DR

This paper establishes a precise link between the arithmetic height of Heegner cycles on arithmetic Shimura curves and the second derivative of a half-integral weight Eisenstein series at s = 1/2, a non-central point in the functional equation. It proves that the generating series of arithmetic heights equals the derivative of an Eisenstein series of weight 3/2 for SL2(Z), with a constant shift, revealing a novel geometric interpretation of the Laurent expansion's second term in terms of Faltings heights of CM abelian surfaces.

ABSTRACT

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated to a Shimura curve and the second term in the Laurent expansion at s=1/2 of an Eisenstein series of weight 3/2 for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s=1/2 is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s=1/2 coincides with the generating function for the degrees of the Heegner cycles on the generic fiber, and, in particular, does not vanish.

Motivation & Objective

• To establish a correspondence between arithmetic heights of Heegner cycles on arithmetic Shimura curves and the second term in the Laurent expansion of a half-integral weight Eisenstein series.
• To demonstrate that the generating series of arithmetic heights arises from the derivative of an Eisenstein series at s = 1/2, a non-central point in the functional equation.
• To provide a geometric interpretation of the non-vanishing first term in the Laurent expansion at s = 1/2 via Faltings heights of CM abelian surfaces.
• To extend the known correspondence between Fourier coefficients and arithmetic invariants to the case where the first term in the Laurent expansion is non-zero.

Proposed method

• Define arithmetic Chow classes  ˆZ(m,v) on the arithmetic surface M associated to a Shimura curve, combining algebraic cycles Z(m) and Green's functions Ξ(m,v).
• Equip the Hodge bundle ω with a canonical metric ||·|| using the holomorphic 1-forms on the universal abelian variety over M.
• Form the height generating series φheight(τ) = ∑⟨ˆZ(m,v), ˆω⟩qm via the Gillet–Soulé height pairing.
• Construct a family of Eisenstein series E(τ,s;D) of weight 3/2 for SL2(Z), with D = D(B) the discriminant of a quaternion algebra B.
• Compute the Laurent expansion of E(τ,s;D(B)) at s = 1/2 and identify the second derivative E′(τ,1/2;D(B)) as the generating series of arithmetic heights.
• Prove the main identity φheight(τ) = E′(τ,1/2;D(B)) + c via direct computation of Fourier coefficients, involving class numbers, L-functions, and logarithmic terms.

Experimental results

Research questions

• RQ1 What is the arithmetic significance of the second term in the Laurent expansion of a half-integral weight Eisenstein series at s = 1/2, a non-central point in the functional equation?
• RQ2 How do the heights of Heegner cycles on arithmetic Shimura curves relate to the Fourier coefficients of Eisenstein series?
• RQ3 Can the non-vanishing first term in the Laurent expansion of an Eisenstein series at s = 1/2 be geometrically interpreted in terms of Faltings heights of CM abelian surfaces?
• RQ4 Is there a general pattern linking arithmetic invariants of Shimura varieties to the Laurent expansions of Eisenstein series, even when the first term is non-zero?

Key findings

• The generating series of arithmetic heights of Heegner cycles on the arithmetic Shimura curve M is equal to the derivative of the Eisenstein series E(τ,s;D(B)) at s = 1/2, up to a constant shift.
• The coefficient of qm in E′(τ,1/2;D(B)) for m > 0 is given by a complex expression involving δ(d;D(B)), H0(m;D(B)), log(d), L′(1,χd)/L(1,χd), and logarithmic terms from primes dividing or not dividing D(B).
• The leading terms in the coefficient expression match the Faltings height of a product of CM elliptic curves Ed × Ed, with h∗Fal(Ed × Ed) = 1/2 log(d) + L′(1,χd)/L(1,χd) − 1/2 log(π) − 1/2γ.
• For m < 0, the coefficient of qm in the derivative is non-zero and involves an integral over r > 1, which is expressed via the function Ψℓ(s,z) and contributes to the height pairing.
• The functional equation E(τ,s;D) = E(τ,−s;D) is established for the renormalized Eisenstein series, relying on the functional equation of L-functions and the doubling formula for the Gamma function.
• The constant term in the Fourier expansion of E(τ,s;D) is shown to be proportional to GD(s) + GD(−s), with GD(s) involving the completed L-function and local factors at primes dividing D.