Superspecial Abelian varieties and the Eichler basis problem for Hilbert modular forms. (English) Zbl 1214.11076
Summary: Let \(p\) be an unramified prime in a totally real field \(L\) such that \(h^{+}(L)=1\). Our main result shows that Hilbert modular newforms of parallel weight two for \(\Gamma _{0}(p)\) can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This may be viewed as a geometric reinterpretation of the Eichler basis problem for Hilbert modular forms.
MSC:
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
Keywords:
superspecial Abelian varieties; Eichler basis problem; Hilbert modular forms; theta series; Eichler ordersReferences:
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