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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
##### Keywords:
Rankin-Selberg $$L$$-function; subconvexity; Heegner points
Zbl 1035.11018
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