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Bounds for automorphic \(L\)-functions. III. (English) Zbl 1163.11325
From the introduction: We continue our study of \(\text{GL}_2\) \(L\)-functions with the aim of providing upper bounds for their order of magnitude. As is familiar it suffices to provide such bounds on the critical line and, both for the sake of applications and for the ideas involved, we are most interested in breaking the convexity bound and this with respect to the conductor. In this paper we are interested primarily in \(L\)-functions attached to characters of the class group of the imaginary quadratic field \(K= \mathbb Q(\sqrt{-D})\). We are motivated by our paper [Duke Math. J. 79, No. 1, 1–56 (1995; Zbl 0838.11058)]. That work was not included in the current series because the class group \(L\)-functions are treated there directly. They may however be viewed as \(L\)-functions associated to cusp forms of weight 1, level \(D\) and character (the nebentypus)
\[ \chi_D(n)= \biggl( \frac{-D}{n}\biggr), \]
the Kronecker symbol (we assume throughout that \(-D\) is a fundamental discriminant).
In this paper we focus on this larger framework and consider \(L\)-functions for cusp forms
\[ f(z)= \sum_l^\infty \lambda_f(n)n^{\frac{k-1}{2}} e(nz)\tag \(*\) \]
of weight \(k\), level \(D\) and any primitive character \(\chi\pmod D\). For a technical reason we assume that \(k\geq 3\) which helps us to resolve easily the convergence problems for various series and integrals (see further comments later). The full space of such cusp forms \(S_k(\Gamma_0(D),\chi_D)\) has a unique finite basis, say \({\mathcal F}\), consisting of primitive forms. To \(f\in{\mathcal F}\) given by the Fourier expansion \((*)\) we attach the \(L\)-function
\[ L(s,f)= \sum_1^\infty \lambda_f(n)n^{-s} \]
and the completed product
\[ \Lambda(s,f)= \biggl( \frac{\sqrt{D}}{2\pi} \biggr)^s\;\Gamma\biggl(s+ \frac{k-1}{2}\biggr) L(s,f). \]
From the Phragmen-Lindelöf principle and the functional equation one easily obtains the “convexity” bound
\[ L(s,f)\ll (k^2|s|^2D)^{\frac14+ \varepsilon}. \]
A classical example of breaking the convexity bound in the conductor aspect is due to D. A. Burgess [Proc. Lond. Math. Soc., III. Ser. 13, 524–536 (1963; Zbl 0123.04404)]. He showed that for Dirichlet \(L\)-functions with any character \(\chi\pmod D\) one has
\[ L(s,\chi)\ll |s|^A D^{\frac{3}{16}+ \varepsilon} \]
while the convexity bound gives only \(\frac14\) in place of \(\frac{3}{16}\). Burgess found an ingenious method for transforming short character sums into high moments of complete sums for which estimates were available and, in particular, the Riemann hypothesis for curves over a finite field (Weil’s theorem) furnishes a strong bound.
In [J. B. Friedlander and H. Iwaniec, Mich. Math. J. 39, 153–159 (1992; Zbl 0765.11037)] an alternative method was given for breaking the convexity barrier. Although it produced a quantitatively weaker result in the Burgess case, it turned out to be possible to apply this new method more generally, in particular, \(\text{GL}_2\) automorphic \(L\)-functions in various aspects. For example in [Invent. Math. 140, No. 1, 227–242 (2000; Zbl 1056.11504)] we proved that
\[ L(s,f)\ll D^{\frac14-\frac{1}{192}+ \varepsilon} \]
for \(f\in S_k(\Gamma_0(D))\), \(k\) even, \(\operatorname{Re}s= \frac12\), \(\varepsilon>0\), the implied constant depending on \(k\), \(s\) and \(\varepsilon\).
In this paper we treat the analogous problem when the cusp form \(f\in S_k(\Gamma_0(D),\chi)\) transforms in accordance with a multiplier given by a primitive character \(\chi\) of conductor equal to the level \(D\). Our first result is
Theorem 1. Let \(k\geq 3\) and \(D\) squarefree. Let \(\chi\pmod D\) be a primitive character with \(\chi(-1)= (-1)^k\). Let \({\mathcal F}\) be the Hecke basis of \(S_k(\Gamma_0(D),\chi)\) and \(L(s,f)\) the \(L\)-function associated to \(f\in{\mathcal F}\). Let \({\mathbf c}= (c_\ell)\) be any sequence of complex numbers with \(c_\ell=0\) if \(\ell\) has a prime divisor \(<z\). Then for \(\operatorname{Re}s=\frac12\) we have
\[ \sum_{f\in{\mathcal F}} \Bigg| \sum_{\ell\leq L} c_\ell\lambda_f(\ell)\Bigg|^2 |L(s,f)|^4\ll \big(\|{\mathbf c}\|^2+ \|{\mathbf c}\|_1^2 z^{-1}\big) |s|^6 D^{1+\varepsilon} \]
for
\[ L=D^\alpha \quad\text{with}\quad \alpha=1/13(48)^2= .0000333\dots \]
where \(\|{\mathbf c}\|\) denotes the \(\ell_2\)-norm and \(\|{\mathbf c}\|_1\) denotes the \(\ell_1\)-norm, the implied constant depending on \(\varepsilon\) and \(k\).
We also obtain
Theorem 2. Let \(k\geq 3\), \(D\) squarefree and \(\chi\pmod D\) a primitive character with \(\chi(-1)= (-1)^k\). Then, for any Hecke cusp form \(f\in S_k(\Gamma_0(D),\chi)\), and \(\operatorname{Re}s=\frac12\), we have
\[ L(s,f)\ll |s|^2 D^{\frac14-\alpha} \]
with \(\alpha=1/2^{18}\) and the implied constant depends only on \(k\).
Theorem 3. Let \(\psi\) be a Hecke character for \(K=\mathbb Q(\sqrt{-d})\) as above. Then for \(\operatorname{Re}s=\frac12\)
\[ L(s,\psi)\ll |s|^2D^{\frac14-\alpha} \]
with \(\alpha=1/2^{18}\) and where the implied constant depends only on \(k\).

MSC:
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F30 Fourier coefficients of automorphic forms
11N36 Applications of sieve methods
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