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Algebraic twists of modular forms and Hecke orbits. (English) Zbl 1344.11036

This is a very interesting and substantial paper that combines analytic number theory, algebraic geometry and automorphic forms. The basic problem is to estimate sums of the type \[ S(f; K; p) := \sum_{n } \lambda_f(n) K(n)V(n/x) \] where \(\lambda(n)\) are Hecke eigenvalues (including divisor-type functions) of a \(\mathrm{GL}(2)\) automorphic form \(f\), \(V\) is a fixed smooth function of compact support, \(K(n)\) is a \(p\)-periodic function “of algebraic origin” and \(x\) is roughly of size \(p\) or a little smaller. The simplest examples for \(K(n)\) are additive or multiplicative characters or more generally normalized exponential sums like (hyper-)Kloosterman sums. In such a situation Poisson/Voronoi summation would be useless to bound \(S(f; K; p)\).
Instead, the authors use a strategy that was first implemented by V. A. Bykovskiĭ [J. Math. Sci., New York 89, No. 1, 915–932 (1998); translation from Zap. Nauchn. Semin. POMI 226, 14–36 (1996) (1996; Zbl 0898.11017)] and generalized by J. B. Conrey and the second author [Ann. Math. (2) 151, No. 3, 1175–1216 (2000; Zbl 0973.11056)] and the reviewer and G. Harcos [J. Reine Angew. Math. 621, 53–79 (2008; Zbl 1193.11044); addendum ibid. 694, 241–244 (2014)]. The form \(f\) of fixed level \(N\), say, is embedded into a spectrally complete family of level \(Np\), and one considers an amplified second moment \[ \sum_g |A(g)|^2 |S(g; K; p)|^2, \] where \(A\) is a suitable amplifier. Opening the square, one first evaluates the spectral sum using the Petersson/Kuznetsov formula, and then the two \(n\)-sums by Poisson summation. The crucial observation that is already implicit in Bykovskiĭ in the case of \(K(n) = \chi(n)\) is that this leads to correlation sums of the type \[ \mathcal{C}(K, \gamma) := \sum_{z \pmod{p}}\hat{K}(\gamma z) \hat{K}(z), \] where \(\hat{K}\) is the mod \(p\) Fourier transform of \(K\) and \(\gamma\) is a matrix in \(\mathrm{PGL}(2, \mathbb{F}_p)\) acting on \(z\) by fractional linear transformations. If \(K\) is such that \(\mathcal{C}(K, \gamma)\) exhibits square-root cancellation for most \(\gamma\), then one can obtain a Burgess-type saving of the form \[ S(f; K; p) \ll p^{7/8 + \varepsilon}. \] The authors show that this is the case for a large class of functions \(K(n)\), given as follows. Let \(\ell \not= p\) be an auxiliary prime, let \(\mathcal{F}\) be an \(\ell\)-adic isotypic trace sheaf on \(\mathbb{A}^{1}_{\mathbb{F}_p}\), as defined in [N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups. Princeton, NJ: Princeton University Press (1988; Zbl 0675.14004)], and fix an isomorphism \(\iota : \bar{\mathbb{Q}}_{\ell} \rightarrow \mathbb{C}\). Then \(K(x) := \iota(\mathrm{tr}(\mathcal{F})(\mathbb{F}_p, x))\) is an admissible function. This does not include additive characters \(K(n) = e(an/p)\), but in this case one has much stronger results by Wilton’s bound, at least if \(f\) is cuspidal.
An important step in the proof is to show that for a trace function \(K\), only a very special class of matrices \(\gamma\) (namely those in the so-called Fourier-Möbius group) can lead to correlation sums without square-root cancellation. This is deduced eventually from the Riemann Hypothesis over finite fields in its most general form. A trace function comes with a conductor (defined in terms of its singularities) that measures the complexity of such a function, and the final bound for \(S(f; K; p)\) depends in terms of \(K\) only on the conductor of \(K\).
Several examples and applications are given. Choosing \(K(n) = \chi(n)\), one recovers a Burgess-type subconvexity bound for twisted \(L\)-functions. Another application concerns algebraic twists of Hecke operators. For a (large) prime \(p\), all but one of the \(p+1\) Hecke points lie on a horocycle of height \(1/p\). It is well known that these points become equidistributed on the modular curve, i.e., the normalized sum of the \(p\) point measures converges to the hyperbolic measure. The authors conclude from their main result that this remains true if the measure is twisted by an isotopic trace function and restricted to an interval of length at least \(p^{7/8 + \varepsilon}\).
The results of this paper are used in several follow-up papers, for instance [the first author et al., Duke Math. J. 163, No. 9, 1683–1736 (2014; Zbl 1318.11103)].

MSC:

11F11 Holomorphic modular forms of integral weight
11F32 Modular correspondences, etc.
11F37 Forms of half-integer weight; nonholomorphic modular forms
11T23 Exponential sums
11L05 Gauss and Kloosterman sums; generalizations
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References:

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