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The \(p\)-adic \(L\)-function for half-integral weight modular forms. (English) Zbl 1470.11099

Summary: The \(p\)-adic \(L\)-function for modular forms of integral weight is well-known. For certain weights the \(p\)-adic \(L\)-function for modular forms of half-integral weight is also known to exist, via a correspondence, established by Shimura, between them and forms of integral weight. However, we construct it here without any recourse to the Shimura correspondence, allowing us to establish its existence for all weights, including those exempt from the Shimura correspondence. We do this by employing the Rankin-Selberg method, and proving explicit \(p\)-adic congruences in the resultant Rankin-Selberg expression.

MSC:

11F37 Forms of half-integer weight; nonholomorphic modular forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F85 \(p\)-adic theory, local fields
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