Special hypergeometric motives and their \(L\)-functions: Asai recognition. (English) Zbl 1511.11047
The generalized hypergeometric functions play an important role in arithmetic and algebraic geometry since they come quite naturally as periods of certain algebraic varieties, and consequently they encode important information about the invariants of these varieties. In the paper under review, the authors explain similar hypergeometric Ramanujan-type formulas for \(\frac{1}{\pi^2}\) in higher rank. Their main result is to experimentally identify that the \(L\)-function of certain specializations of hypergeometric motives araising from these formulas can be derived from Asai \(L\)-functions of Hilbert modular forms of weight \((2, 4)\) over real quadratic fields.
Reviewer: Sami Omar (Sukhair)
MSC:
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 33F10 | Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) |
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