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Nelson, Paul D.

Subconvex equidistribution of cusp forms: reduction to Eisenstein observables. (English) Zbl 1428.11093

Duke Math. J. 168, No. 9, 1665-1722 (2019).
The subconvexity problems for families of the automorphic \(L\)-functions are closely related to important arithmetical equidistribution problems. In the paper under the review, the author shows that, under certain assumptions, the subconvexity problem for the family \(L(\mathrm{ad}(\pi) \otimes \tau, \frac{1}{2})\), with \(\tau\) on \(\mathrm{PGL}_2\) fixed and \(\pi\) of \(\mathrm{GL}_2\) varying, can be reduced to the subconvexity problem for \(L(\pi \otimes \overline{\pi} \otimes \chi, \frac{1}{2})\), with \(\chi\) on \(\mathrm{GL}_1\) fixed and \(\pi\) on \(\mathrm{GL}_2\) varying.
More precisely, if \(\pi\) is a sequence of cuspidal automorphic representations of \(\mathrm{GL}_2\) with unramified central character, large prime and bounded infinity type, then the subconvexity for \(\mathrm{GL}_1\)-twists of the adjoint \(L\)-function of \(\pi\) implies the subconvexity for \(\mathrm{PGL}_2\)-twists of the adjoint \(L\)-function of \(\pi\) (with polynomial dependence upon the conductor of the twists).
Reviewer: Ivan Matić (Osijek)

Cited in 1 Review
Cited in 11 Documents

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F27 Theta series; Weil representation; theta correspondences
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity

Keywords:

subconvexity; \(L\)-functions; automorphic forms; Eisenstein series
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References:

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