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Uniform subconvexity and symmetry breaking reciprocity. (English) Zbl 1492.11098

The main result of this paper is the new subconvexity result for \(\mathrm{GL}(2)\). It provides new uniform subconvexity bounds for \(L\)-functions in the level and the eigenvalue aspect. More precisely, let \(q\) be a prime, \(\tau\in\mathbb{R}\), \(t\in\mathbb{R}\cup [-i\vartheta_0,i\vartheta_0]\), and \(T=1+|t|\), where \(\vartheta_0\leq 7/64\) denotes an admissible exponent for the Selberg eigenvalue conjecture. Let \(f\) be a Hecke-Maaß cusp form of eigenvalue \(\lambda =1/4 + t^2\), level \(q\) and primitive central character. Then \[ L(1/2+i\tau,f)\ll_{\tau,\varepsilon} (qT)^\varepsilon \left( q^{\frac{1}{4}-\frac{1}{128}}T^{\frac{1}{2}-\frac{1-2\vartheta_0}{20}} + q^{\frac{1}{8}}T^{\frac{1}{2}} \right) \] for any \(\varepsilon >0\). This improves the previous subconvexity results in the level aspect obtained by W. Duke et al. [Invent. Math. 149, No. 3, 489–577 (2002; Zbl 1056.11072)] and by V. Blomer et al. [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 5, 697–740 (2007; Zbl 1185.11034)]. Although the same result with the same proof would hold for holomorphic forms, with the weight playing the role of \(T\), this paper puts focus on Maaß forms, because the case of holomorphic forms is considerably easier [W. Duke et al., Invent. Math. 143, No. 2, 221–248 (2001; Zbl 1163.11325)].
Perhaps even more important then the result itself is the symmetry breaking reciprocity formula, which is the key ingredient in its proof. It is a reciprocity formula relating
the fourth moment of automorphic \(L\)-functions of level \(q\) an primitive central character twisted by \(\ell\)-th Hecke eigenvalue, that is, \[ \sum_{f\in \mathcal{B}(q,\chi)} |L(1/2,f)|^4 \lambda_f(\ell), \] where the sum is over an orthogonal basis for the space of cuspidal Maaß newforms of level \(q\) and a primitive character \(\chi\) modulo \(q\), and \(\lambda_f(\ell)\) is the \(\ell\)-th Hecke eigenvalue of \(f\), and
the twisted mixed moment of automorphic \(L\)-functions of level \(\ell\) and trivial central character, that is, \[ \sum_{f\in \mathcal{B}(\ell,\mathrm{triv})} L(1/2,f)^3 L(1/2,f\times\bar{\chi}), \] where the sum is over an orthogonal basis for the space of cuspidal Maa{ss} forms of level \(\ell\) and the trivial central character, where \(\ell\) is prime.
The point is that the \(L\)-functions in this reciprocity formula are not of the same type on both sides of the identity.
Reviewer: Neven Grbac (Pula)

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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References:

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