Sparse equidistribution problems, period bounds, and subconvexity

Abstract

We introduce a ``geometric'' method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier coefficients of automorphic forms.

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Cited by in corpus (9)

• The subconvexity problem for $\GL_{2}$
• Distribution of periodic torus orbits and Duke's theorem for cubic fields
• Effective equidistribution of translates of large submanifolds in semisimple homogeneous spaces
• Heegner points and Eisenstein series
• Test vectors for trilinear forms : the case of two principal series
• Test vectors for trilinear forms, when two representations are unramified
• Test vectors for trilinear forms, when two representations are unramified and one is special
• Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces
• Duke's Theorem and Continued Fractions