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The subconvexity problem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points. (English) Zbl 1068.11033
The author considers Rankin-Selberg \(L\)-functions \(L(f\otimes g,s)\), where \(f\) and \(g\) are primitive cusp forms with \(g\) fixed and holomorphic of weight \(\geq 1\). Let \(f\) and \(g\) be of level \(q\), \(D\) and of Nebentypus \(\chi _f\), \(\chi _g\), respectively, and suppose that \(\chi _f\chi _g\) is not trivial. The main result of the paper is the estimate \(L^{(j)}(f\otimes g,s) \ll q ^{1/2-1/1057}\), where \(\operatorname{Re} s =1/2\) and the implied constant depends on \(j\), \(s\), \(D\) and of the weight or spectral parameter of \(f\) depending on whether \(f\) is holomorphic or real analytic. The point here is the subconvexity of the estimate as compared with the convexity bound \(\ll q^{1/2+\varepsilon }\).
This is a generalization of an earlier result due to E. Kowalski, P. Michel and J. Vanderkam [Duke Math. J. 114, 123–191 (2002; Zbl 1035.11018)]. The proof is very complicated and ingenious, being based on the amplification technique. The equidistribution of certain Heegner points is discussed as an application.

MSC:
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11G05 Elliptic curves over global fields
Citations:
Zbl 1035.11018
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