The spectral decomposition of shifted convolution sums over number fields

Maga, Péter (2017) The spectral decomposition of shifted convolution sums over number fields. JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. pp. 1-22. ISSN 0075-4102

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Abstract

Let π1, π2 be cuspidal automorphic representations of GL2 over a number field F with Hecke eigenvalues λπ1(m),λπ2(m). For nonzero integers l1,l2∈F and compactly supported functions W1,W2 on F×∞, a spectral decomposition of the shifted convolution sum ∑l1t1−l2t2=q0≠t1,t2∈nλπ1(t1n−1)λπ2(t2n−1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯N(t1t2n−2)√W1(l1t1)W2(l2t2)¯¯¯¯¯¯¯¯¯¯¯¯¯ is obtained for any nonzero fractional ideal n and any nonzero element q∈n.

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