Weyl-type hybrid subconvexity bounds for twisted LL-functions and Heegner points on shrinking sets

  • Matthew P. Young

    Texas A&M University, College Station, USA
Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets cover
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Abstract

Let qq be odd and squarefree, and let χq\chi_q be the quadratic Dirichlet character of conductor qq. Let uju_j be a Hecke–Maass cusp form on Γ0(q)\Gamma_0(q) with spectral parameter tjt_j. By an extension of work of Conrey and Iwaniec, we show L(uj×χq,1/2)ε(q(1+tj))1/3+εL(u_j \times \chi_q, 1/2) \ll_{\varepsilon} (q (1 + |t_j|))^{1/3 + \varepsilon}, uniformly in both qq and tjt_j. A similar bound holds for twists of a holomorphic Hecke cusp form of large weight kk. Furthermore, we show that L(1/2+it,χq)ε((1+t)q)1/6+ε|L(1/2+it, \chi_q)| \ll_{\varepsilon} ((1 + |t|) q)^{1/6 + \varepsilon}, 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.

Cite this article

Matthew P. Young, Weyl-type hybrid subconvexity bounds for twisted LL-functions and Heegner points on shrinking sets. J. Eur. Math. Soc. 19 (2017), no. 5, pp. 1545–1576

DOI 10.4171/JEMS/699