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Bounds for automorphic $$L$$-functions. (English) Zbl 0765.11038
Consider a Dirichlet character $$\chi$$ of modulus $$q$$ and the automorphic $$L$$-function obtained on twisting by $$\chi$$ an arbitrary holomorphic cusp form $$f$$ of weight $$k$$ for the full modular group (although it is stated by the authors that the method of the paper applies more generally). Such an $$L$$-function satisfies a functional equation, proven by Hecke, and from this and the Phragmen-Lindelöf convexity principle one obtains an upper bound for the order of magnitude of the $$L$$-function on the critical line (which may by normalizing be taken to be $$\text{Re }s=1/2$$) of the form $$L(s)\ll q^ \alpha$$ for every $$\alpha>1/2$$. (Here and below we ignore the dependence on $$s$$ and also improvements of logarithmic type in the $$q$$ aspect.) The authors give the stronger estimate wherein, in the exponent, 1/2 is replaced by 5/11. The proof combines the ideas of the paper of J. Friedlander and H. Iwaniec (see the preceding review) with the ideas developed in a series of papers by W. Duke and H. Iwaniec, “Estimates for coefficients of $$L$$-functions I- IV” [see I: Automorphic forms and analytic number theory, Proc. Conf., Montreal/Can. 1989, 43-47 (1990; Zbl 0745.11030); II: Proc. Amalfi Conf. Analytic Number Theory, 1989 Maiori/Salerno, 71-82 (1992; Zbl 0787.11020); III: Prog. Math. 102, 113-120 (1992; Zbl 0763.11024); IV: Am. J. Math. 116, 207-217 (1994; Zbl 0820.11032) and the references therein].
An even stronger bound, namely with exponent 3/7, but at the special point $$s=1/2$$ only, follows from an earlier argument due to H. Iwaniec [Invent. Math. 87, 385-401 (1987; Zbl 0606.10017)], deduced via Waldspurger’s theorem from an estimate for Fourier coefficients of the half-integral weight form related to $$f$$ under the Shimura correspondence. This latter estimate was used by W. Duke [Invent. Math. 92, 73-90 (1988; Zbl 0628.10029)]to give a resolution of the Linnik problem and the equi-distribution on integer points of the sphere. By using again the Waldspurger correspondence, but this time in the opposite direction, the authors deduce an estimate for the relevant Fourier coefficient which is sufficient to give a new proof of the equi- distribution theorem.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11L40 Estimates on character sums 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11F30 Fourier coefficients of automorphic forms
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##### References:
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