Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets
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© European Mathematical Society 2017. Let q be odd and squarefree, and let Xq be the quadratic Dirichlet character of conductor q. Let uj be a Hecke-Maass cusp form on Γ0(q) with spectral parameter tj . By an extension of work of Conrey and Iwaniec, we show L(uj × q ; 1/2) ≪ϵ (q(1 + |tj|))1/3+ϵ, uniformly in both q and tj . A similar bound holds for twists of a holomorphic Hecke cusp form of large weight k. Furthermore, we show that |L(1/2 + it; Xq)| ≪ϵ ((1 + |t|/q)1/6+ϵ, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.
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