Michael Rapoport Mathematisches Institut der Universität zu Köln Weyertal 86-90 D - 50931 Köln email: rapoport@mi.uni-koeln.de

Intersection numbers of Heegner cycles

This is a report on work done to put into reality Kudla’s dream which is to relate the central value of the derivative of certain Siegel-Eisenstein series of genus n with the intersection pairing of special cycles (Heegner cycles and their generalizations) on arithmetic models of Shimura varieties associated to orthogonal groups of signature (n − 1, 2). In this abstract I will concentrate on my joint results with S.Kudla on the case n = 3. In this case one is considering arithmetic Hirzebruch-Zagier cycles on arithmetic Hilbert-Blumenthal surfaces. Note that the choice of topic is very appropriate for a talk in Bonn, where this theme has a long tradition.

Let V be a quadratic space of signature (2, 2) over Q. Then B = C^{+}(V) is an indefinite quaternion algebra over a real quadratic field k. Let

G = {g ∈ B^{×}; gg^{ι} ∈ Q^{×}} as algebraic group over Q .

Then G acts on V ⊂ C^{−}(V) by conjugation. Let

D = space of oriented negative 2-planes in V (R).

The corresponding Shimura variety (a finite disjoint sum of classical Hilbert-Blumenthal surfaces)

Sh(G, D) = G(Q) \ [D × G(A_{f})/K]

is defined over Q as the moduli variety M parametrizing 4-tuples (A, λ, ι, \overline{η}) consisting (roughly speaking) of an abelian variety of dimension 8 up to isogeny, a polarization of it, an action of C(V) ⊗_{Q} k on it, and a level-K-structure. A special endomorphism of (A, λ, ι, \overline{η}) is an element j of End(A) such that

j ∘ ι(c ⊗ a) = ι(c ⊗ a′) ∘ j
j^{∗} = j  (Rosati-involution)

Since j^{2} = Q(j)·id_{A} for special j, the special endomorphisms of (A, λ, ι, \overline{η}) form a quadratic space over Q. For n > 0 and T ∈ Sym_{n}(Q)_{≥0} and a K-invariant compact open subset ω ⊂ V (A_{f})^{n} we obtain the moduli space Z(T, ω) over M whose fibre over (A, λ, ι, \overline{η}) parametrizes n-tuples j = (j_{1}, …, j_{n}) of special endomorphisms with Q(j) = T and satisfying an integrality condition depending on ω. For n = 1, B = M_{2}(k) and suitable choices of T ∈ Q_{>0} and ω we obtain for Z(T, ω) the classical Hirzebruch-Zagier cycles T_{N}.

The scheme M can be extended from Spec Q to Spec Z[1/N] for sufficiently big N (which one can specify), so as to obtain a smooth quasi-projective scheme M of relative dimension 2 over Spec Z[1/N]. Similarly, for T ∈ Sym_{n}(Z[1/N])_{≥0} and ω ⊂ V (A_{f})^{n} satisfying an integrality condition outside N we obtain an arithmetic cycle Z(T, ω) on M.

Theorem 1: Let T ∈ Sym_{3}(Z[1/N])_{≥0}. If det(T) ≠ 0, then Z(T, ω) has empty generic fibre. If furthermore Z(T, ω) ≠ ∅, then it is concentrated in characteristic p for a unique p and in fact is contained in the supersingular locus of M ×_{Spec Z[1/N]} Spec F_{p}.

One might expect that the cycle Z(T, ω) in the arithmetic 3-fold M should be a discrete set of points, for T ∈ Sym_{3}(Z[1/N])_{>0}. However, this is not true in general.

Theorem 2: Let T ∈ Sym_{3}(Z[1/N])_{>0} and let Z(T, ω)(\overline{F}_{p}) ≠ ∅. Then dim(Z(T, ω)) = 0, iff p splits in k or p is inert in k and T ≠ 0 (mod p).

If T ≡ 0 (mod p) for inert p, then Z(T, ω) is a union of certain of the projective lines that make up the supersingular locus in M ×_{Spec Z[1/N]} Spec F_{p}, and these can be enumerated in terms of the Bruhat-Tits tree of P GL_{2}(Q_{p^{2}}).

We form a generating series

(∗)  ∑_{T} deg(Z(T, ω))q^{T} .

Here T runs over the matrices characterized in Theorem 2 and deg Z = log #(O_{Z}).

Theorem 3: Consider the Siegel modular form of genus 3 and weight 2,

ϕ(τ) = det(v)^{−1} · E′(h_{τ} , 0, Φ) , τ = u + iv, h_{τ} (i) = τ ,

where Φ depends on ω. Then the sum (∗) is part of the Fourier expansion of ϕ,

ϕ(τ) = ∑_{T ∈Sym_{3}(Z[1/N])} a_{T} (v)q^{T} , q^{T} = e^{2πi tr(T τ)} .

The proof of the last theorem is by explicitly computing the coefficients of both series. For the coefficients of (∗) this is done by reducing it to a problem of deforming a homomorphism between two supersingular elliptic curves. This problem had been solved by Gross and Keating. For the Fourier coefficients of ϕ(τ) one uses results of Kitaoka on local representation densities of quadratic forms.

The problem of intersecting arithmetic special cycles on M may be (partially) reduced to the previous three theorems by the following remark. Let n_{1}, …, n_{r} be positive integers with n_{1} + … + n_{r} = 3. For each i = 1, …, r let T_{i} ∈ Sym_{n_{i}}(Z[1/N])_{>0} and ω_{i} ⊂ V (A_{f})^{n_{i}} as before, so that we obtain the special cycle Z(T_{i}, ω_{i}) on M. Then

Z(T_{1}, ω_{1}) ×_{M} … ×_{M} Z(T_{r}, ω_{r}) = ∐_{T} Z(T, ω) .

Here on the index runs over T ∈ Sym_{3}(Z[1/N])_{≥0} with diagonal blocks T_{1}, …, T_{r}.

For more details the reader can consult

S.Kudla, M.Rapoport: Arithmetic Hirzebruch-Zagier cycles. alg-geom 9904083.