Zero-cycles on Hilbert-Blumenthal surfaces
Let \(F\) be a real quadratic field of class number 1 and discriminant \(q\), and let \(X/{\mathbb{Q}}\) be a smooth complete variety obtained from the coarse moduli space parameterizing polarized abelian surfaces with real multiplication by the ring of integers \(O_F\), obtained by resolving the cusps and finite quotient singularities. One may extend \(X\) to a smooth scheme over \({\mathbb{Z}}_p\) for \(p\in U\) - a set containing all but finitely many primes \(p\). The author makes the very restrictive hypothesis that the Doi-Naganuma lift \(S_2(\Gamma_0(q), \varepsilon_q) \rightarrow S_2(\text{SL}_2(O_F))\) is surjective. Under the hypotheses above the author proves some very strong theorems concerning zero-cycles on these varieties, the Tate conjecture for divisors on the reduction modulo \(p\) and more. In particular: Theorem A: Let \(p\) be a prime that splits in \(F\) and such that \(X\) has good reduction at \(p\) with closed fiber \(X_p\). Then the Tate conjecture holds for divisors on \(X_p\). That is: \(NS(X_p)\otimes {\mathbb{Q}}_\ell \cong H^2_{et}(\overline{X_p}, {\mathbb{Q}}_\ell(1))^{Fr_p}\). Theorem B: The boundary map \(\partial ': H^1(X, K_2)\otimes \mathbb{Q} \rightarrow \oplus_\ell \text{Pic}(X_\ell) \otimes \mathbb{Q}\) is surjective, where the summation on the r.h.s. is over all primes \(\ell \in U\) that are split in \(F\). Theorem C: For all split primes \(p\in U\) the prime to \(p\) torsion in \(\text{Ch}_0(X\times_{{\mathbb{Z}}_p} {\mathbb{Q}}_p)\) is finite. Under additional hypothesis involving Beilinson's conjectures, Tate's conjecture at inert primes and more, the author obtains the following Theorem D: For all \(p\in U-\{2, 3\}\) the \(p\)-torsion part of \(\text{Ch}_0(X)\) is finite. The proof of the Tate conjecture is based on reduction to an analogous statement for the intersection cohomology, which, in turn, is studied via modular forms. In this study the assumption on the Doi-Naganuma lift is used to reduce the study of Galois modules, a priori associated to Hilbert modular forms, to Galois modules associated to elliptic modular forms.
scientific article; zbMATH DE number 1463082
scientific article