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Shifted convolution sums for \(\mathrm{GL}(3)\times\mathrm{GL}(2)\). (English) Zbl 1330.11033

From the text: For the shifted convolution sum \[ D_h(X)=\sum_{m=1}^\infty {\lambda}_1(1,m){\lambda}_2(m+h)V\left(\frac{m}{X}\right), \] where \({\lambda}_1(1,m)\) are the Fourier coefficients of an \(\mathrm{SL}(3,\mathbb Z)\) Maass form \({\pi}_1, {\lambda}_2(m)\) are those of an \(\mathrm{SL}(2,\mathbb Z)\) Maass or holomorphic form \({\pi}_2\), and \(1\leq |h|\ll X^{1+\varepsilon}\), we establish the bound \[ D_h(X)\ll _{\pi_1,\pi_2,\varepsilon} X^{1-1/20+\varepsilon}. \] The bound is uniform with respect to the shift \(h\).
As in the case of the \(\mathrm{GL}(2)\) shifted convolution sum we first apply the circle method to detect the shift using additive harmonics, and then apply Voronoi summation formula. However, unlike the \(\mathrm{GL}(2)\) case, this does not solve the problem. We are left with a complicated expression, involving higher dimensional Kloosterman-type character sums. Assuming square root cancellation in the character sum one can show that we are just at the threshold, and any saving in the sum of the character sums will yield a non-trivial bound. However, except in the case of the zero shift \(h = 0\), it is not clear how to obtain extra cancellation. We resolve this issue by adopting Jutila’s variation of the circle method with an important new input – factorizable moduli. This seemingly simple idea has other important applications. In [“The circle method and bounds for \(L\)-function.” (Preprint)] we apply this idea to several subconvexity problems.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11M41 Other Dirichlet series and zeta functions

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References:

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