[Paper Review] Special cycles and derivatives of Eisenstein series

Stephen S. Kudla|ArXiv.org|Aug 29, 2003

Advanced Algebra and Geometry33 references4 citations

TL;DR

This paper proposes a conjectural framework linking algebraic cycles on orthogonal Shimura varieties, derivatives of Eisenstein series, and special values of Rankin-Selberg L-functions. It establishes a geometric realization of the Rallis inner product formula via theta lifts, showing that cup products of cohomology classes from Siegel modular forms arise from pullbacks of Eisenstein series, with key identities linking cohomology pairings to L-values and volumes of special cycles.

ABSTRACT

This article sketches relations among algebraic cycles for the Shimura varieties defined by arithmetic quotients of symmetric domains for O(n,2), theta functions, values and derivatives of Eisenstein series and values and derivatives of certain L-functions. In the geometric case, results of joint work with John Millson imply that the generating functions for the classes in cohomology of certain algebraic cycles of codimension r are Siegel modular forms of genus r and weight n/2+1. A result of Borcherds shows that, for r=1, the same is true for the generating function for the classes of such divisors in the Chow group. By the Siegel-Weil formula, the generating function for the volumes of codimension r cycles coincides with a value of a Siegel-Eisenstein series of genus r. In particular, this gives an interpretation of the Fourier coefficients of these Eisenstein series as volumes of algebraic cycles. The second part of the paper discusses the possible analogues of these results in the arithmetic case, where the special values of derivatives of Eisenstein series arise. In this case, the Fourier coefficients of such derivatives are should be the heights (arithmetic volumes) of certain cycles on integral models of the O(n,2) type Shimura varieties. Relations of this sort would yield relations between central derivatives of certain L-functions and height pairings. The case of curves on a Siegel 3-fold and of the central derivative of a triple product L-function are discussed.

Motivation & Objective

• To formulate a broad conjectural framework connecting algebraic cycles, Eisenstein series derivatives, and special values of Rankin-Selberg L-functions.
• To provide a geometric interpretation of the Rallis inner product formula in terms of cohomology classes on Shimura varieties.
• To relate the cup product of cohomology classes associated with Siegel modular forms to the pullback of Eisenstein series.
• To generalize the arithmetic inner product formula to higher-rank classical groups using theta lifts and automorphic forms.
• To establish a bridge between arithmetic geometry and automorphic forms via the cohomological realization of special cycles and L-functions.

Proposed method

• Constructs sub-Shimura varieties Z(x) ⊂ MK as special cycles via rational vectors x ∈ V(Q) with positive norm.
• Uses the Weil representation and theta correspondence to associate Siegel modular forms to cohomology classes in H^{2r}(MK).
• Applies the classical theta lift to define a map from cusp forms f ∈ S(r)_{n/2+1} to cohomology classes [θr(f,ϕ)] ∈ H^{2r}(MK).
• Derives a transformation law for vector-valued theta functions using the action of G’_Q and the metaplectic cover, yielding classical modular transformation properties.
• Computes the cup product of two such cohomology classes via a doubling integral involving Eisenstein series E_{2n}(τ, s, ϕ₁⊗ϕ̄₂).
• Establishes a conjectural identity (A.II.19) expressing the cup product as the pullback of an Eisenstein series to H^{r₁} × H^{r₂}.

Experimental results

Research questions

• RQ1 How are algebraic cycles on orthogonal Shimura varieties related to derivatives of Eisenstein series?
• RQ2 What is the geometric meaning of the Rallis inner product formula in terms of cohomology classes on Shimura varieties?
• RQ3 Can the cup product of cohomology classes arising from Siegel modular forms be expressed as a pullback of an Eisenstein series?
• RQ4 What is the precise relationship between special values of Rankin-Selberg L-functions and the volumes of special cycles?
• RQ5 How do theta lifts and automorphic forms on classical groups realize the conjectural arithmetic inner product formula?

Key findings

• The cup product of cohomology classes [θr(f₁,ϕ₁)] and [θr(f₂,ϕ₂)] is given by the Petersson inner product of f₁⊗f̄₂ with the Eisenstein series E_{2n}(τ, 1/2, ϕ₁⊗ϕ̄₂), up to a constant B.
• When f₁ and f₂ correspond to the same cuspidal automorphic representation π, the pairing yields L(1, π) times the Petersson norm of f, as in (A.II.17).
• The identity (A.II.19) expresses the cup product as the pullback of the Eisenstein series En(τ, 1/2, ϕ₁⊗ϕ̄₂) to H^{r₁} × H^{r₂}, linking cohomology to Eisenstein series.
• For anisotropic V, the volume of MK with respect to the canonical n-form Ω^n satisfies vol(MK, Ω^n) · En(τ, 1/2, ϕ) = ∑_{T≥0} vol(Z(T,ϕ)) q^T, as in (A.II.18).
• The doubling integral in (A.II.14) reduces to a standard L-function via known results from [51], yielding L(1, π) as a key factor.
• The conjectural identity (6.10) is generalized to (A.II.19), showing that the cup product of cohomology classes is realized as a pullback of an Eisenstein series.