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Estimates of automorphic \(L\)-functions in the discriminant-aspect. (English) Zbl 1065.11030

Let \(\phi\) be a Maaß cusp form for the Bianchi group \(\text{PSL}(2,{\mathcal O}_d)\) where \({\mathcal O}_d\) is the ring of integers of the imaginary quadratic number field with discriminant \(d<0\), \(d\neq -3,-4\). Denote the class-number of \({\mathcal O}_d\) by \(h(d)\) and assume that \(\phi\) is an even Hecke eigenform. Then the author’s main theorem gives the following estimate for the \(L\)-function \(L(s,\phi)\) attached to \(\phi\): For any \(\varepsilon>0\) we have \(L(s,\phi)\) attached to \(\phi\): For any \(\varepsilon>0\) we have \[ L\bigl(\tfrac12+it,\phi \bigr)\ll_\varepsilon|\sqrt d(1+|r|)|^{1+\varepsilon h(d)}(1+|t|)^{1+\varepsilon} \] where \(\lambda=1+r^2\) is the eigenvalue for \(\phi\) and where the implied constant depends solely on \(\varepsilon\). The proof rests on an estimate of the Hecke eigenvalues which is of independent interest.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F55 Other groups and their modular and automorphic forms (several variables)
11M41 Other Dirichlet series and zeta functions
32A30 Other generalizations of function theory of one complex variable
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