Zero-cycles on Hilbert-Blumenthal surfaces

Abstract

Introduction. The main object of study in this paper with respect to zero-cycles is a special class of Hilbert-Blumenthal surfaces X, which are defined over Q as smooth compactifications of quasi-projective varieties S/Q, more precisely, of coarse moduli schemes S that represent the moduli stack of polarized abelian surfaces with real multiplication by the ring of integers in a real quadratic field F = Q( √ d). We assume that d = q is a prime ≡ 1(4) and that the class number of F is 1. Then S(C), the complex points of S, can be described as H×H/ SL2(OF ), where H is the upper halfplane. In the early seventies, Hirzebruch and Zagier [HZ] defined for each integer N a curve TN on S (called Hirzebruch-Zagier cycles) and showed that their intersection numbers occur as Fourier coefficients of modular forms of level q with Nebentypes eq , the quadratic character of F/ Q. In this connection with modular forms, HirzebruchZagier cycles reveal very similar properties to Hecke correspondences on the selfproduct of the modular curve X0(q). This crucial observation of Hirzebruch and Zagier, together with Tunnell’s proof of the Tate conjecture for a product of modular curves, inspired Harder, Langlands, and Rapoport [HLR] to prove the Tate conjecture for divisors on Hilbert-Blumenthal surfaces over abelian number fields. (The proof of the Tate conjecture was then accomplished by Klingenberg [Kl] and Murty and Ramakrishnan [MR] in the general case.) From the new strategy to study torsion zero-cycles on algebraic surfaces as developed in [LS] and used in [L1], [L2] (compare also [LR]), it is clear that a crucial point is the Tate conjecture in characteristic p at good reduction primes. As one of our main results, we prove the Tate conjecture in characteristic p for primes p that split in F for a certain class of Hilbert-Blumenthal surfaces. For this we recall that to each modular cusp form f of weight 2, level q, and Nebentypes eq , that is, f ∈ S2(� 0(q), eq ) ,w e associate a Hilbert modular cusp form ˆ f ∈ S2(SL2(OF )) under the Doi-Naganuma