Modular abelian varieties over number fields. (English) Zbl 1294.11096
Suppose that \(K\) is a Galois number field over \(\mathbb{Q}\). Let \(B\) be a non-CM abelian variety over \(K\) satisfying the following two conditions:
1) \(B\) is a variety over \(\mathbb{Q}\) admitting isogenies \(\mu_{\sigma} : {^\sigma}B \to B\) compatible with \(\mathrm{End}(B)\) defined over \(K\) for all \(\sigma\) in the Galois group \(\mathrm{Gal}(K:Q)\) and
2) \(\mathrm{End}_K(B)\) is a division algebra with center being a number field \(E\) such that the index \(t \leq 2\), and the reduced degree \(t[E:\mathbb{Q}] = \mathrm{dim}(B)\). These abelian varieties are called \(K\)-building blocks.
An abelian variety \(B\) over \(K\) is called strongly modular if its \(L\)-function \(L(B/K;s)\) is equivalent to a product of \(L\)-series of newforms over \(\mathbb{Q}\) for congruence subgroups of the form \(\Gamma_1(N)\). The main result of this paper is a characterization of strongly modular abelian varieties \(B\) over \(K\) using \(K\)-building blocks. More precisely, if \(B\) is a \(K\)-simple abelian variety, then \(B\) is strongly modular over \(K\) if and only if \(B\) is a \(K\)-building block that satisfies two extra conditions: 1) \(K\) over \(\mathbb{Q}\) is abelian;
2) the cocycle class \([c_{B/K}]\) is in the subgroup \(\mathrm{Ext}(G,E^*) \subset H^2(G,E^*)\) consisting of symmetric cocycle classes. In the last section, the authors also provide some concrete examples of abelian surfaces with quaternionic multiplication as applications of the general results.
1) \(B\) is a variety over \(\mathbb{Q}\) admitting isogenies \(\mu_{\sigma} : {^\sigma}B \to B\) compatible with \(\mathrm{End}(B)\) defined over \(K\) for all \(\sigma\) in the Galois group \(\mathrm{Gal}(K:Q)\) and
2) \(\mathrm{End}_K(B)\) is a division algebra with center being a number field \(E\) such that the index \(t \leq 2\), and the reduced degree \(t[E:\mathbb{Q}] = \mathrm{dim}(B)\). These abelian varieties are called \(K\)-building blocks.
An abelian variety \(B\) over \(K\) is called strongly modular if its \(L\)-function \(L(B/K;s)\) is equivalent to a product of \(L\)-series of newforms over \(\mathbb{Q}\) for congruence subgroups of the form \(\Gamma_1(N)\). The main result of this paper is a characterization of strongly modular abelian varieties \(B\) over \(K\) using \(K\)-building blocks. More precisely, if \(B\) is a \(K\)-simple abelian variety, then \(B\) is strongly modular over \(K\) if and only if \(B\) is a \(K\)-building block that satisfies two extra conditions: 1) \(K\) over \(\mathbb{Q}\) is abelian;
2) the cocycle class \([c_{B/K}]\) is in the subgroup \(\mathrm{Ext}(G,E^*) \subset H^2(G,E^*)\) consisting of symmetric cocycle classes. In the last section, the authors also provide some concrete examples of abelian surfaces with quaternionic multiplication as applications of the general results.
Reviewer: Xiao Xiao (Utica)
MSC:
11G10 | Abelian varieties of dimension \(> 1\) |
11G18 | Arithmetic aspects of modular and Shimura varieties |
11F11 | Holomorphic modular forms of integral weight |