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The subconvexity problem for Artin \(L\)-functions. (English) Zbl 1056.11072

This article continues and in a sense completes the joint work of the authors from a series of at least 5 articles, the most recent one of which appeared in 2001 [Bounds for automorphic \(L\)-functions. III, Invent. Math. 143, 221-248 (2001; Zbl 1163.11325)].
The word “subconvexity” in the title refers to the general problem of estimating the absolute value of a Dirichlet series (usually arising either from a Dirichlet character, an automorphic form or some Galois representation) with analytic continuation to all of \(\mathbb C\) and functional equation under \(s \mapsto 1-s\) on the critical line \(\operatorname{Re}(s)=1/2\): Under relatively mild conditions on the Dirichlet series at hand the convexity principle of Phragmen and Lindelöf allows to give an estimate depending on data like degree, \(\Gamma\)-factor, conductor of the Dirichlet series or its defining arithmetic data, which is usually referred to as the convexity bound. In the example of the Artin \(L\)-series \(L(s,\rho)\) associated to a Galois representation of conductor \(D\) the assertion that the analytic continuation is possible is the Artin conjecture (established in the case of degree \(2\) unless the representation is icosahedral) and the convexity estimate gives under this assumption \(| L(s,\rho)|\ll_{\varepsilon,s} D^{1/4+\varepsilon}\) for all \(\varepsilon >0\) on the line \(\operatorname{Re}(s)=1/2\). The suspected truth, on the other hand, is the estimate \(| L(s,\rho)|\ll_{\varepsilon,s} D^{\varepsilon}\), which would follow from the Lindelöf hypothesis and hence from the generalized Riemann hypothesis for these \(L\)-functions. Any estimate better than the convexity bound is referred to as a subconvexity estimate (or an estimate breaking the convexity barrier); usually all such convexity breaking estimates have very strong arithmetic corollaries.
The series of articles mentioned is mostly concerned with the case of Galois representations of degree \(2\) (excluding the icosahedral case as long as the Artin conjecture is not proven for this case) where one knows that the \(L\)-function at hand is indeed an automorphic \(L\)-function and can hence use methods from the theory of automorphic forms and their \(L\)-functions. Having settled special cases in their earlier papers the authors prove in the present article the subconvexity estimate \[ | L(s)|\ll(|\lambda|^{1/2}+| s |)^{10}D^{1/4-1/23041} \] for the \(L\)-series of a Hecke-Maaßcusp form of weight \(k\), Laplace eigenvalue \(\lambda\) and character \(\chi\) satisfying \(\chi(-1)=(-1)^k\) for the group \(\Gamma_0(D)\) and get from this the subconvexity estimate \(| L(s,\rho)|\ll| s |^{10}D^{1/4-1/23041}\) for an Artin \(L\) function of degree \(2\) as above (not icosehedral), where the determinant character of \(\rho\) has the same conductor \(D\) as \(\rho\). As expected one gets deep arithmetic results from this; the authors mention in particular the existence of primitive ideals of small norms in all cosets of a (large) subgroup of the ideal class group of a quadratic number field and the generation of cyclic subgroups of the class group by ideals of small norm.
It is rather difficult to describe in a short review all the technical difficulties that the authors have to overcome on the way to their result, a sketch of this can be found in Section 3 of the article. I prefer to mention here only that they use the method usually called the amplification method of Iwaniec; moreover, the authors use a summation formula of Petersson-Kuznetsov type resulting in a rather intricate comparison of spectral data from the relevant spaces of automorphic forms with sums of Kloosterman sums.
It should also be pointed out that for the convenience of the reader the article contains (among others) rather complete summaries of the necessary results and concepts from the spectral theory of automorphic forms for a Hecke congruence subgroup \(\Gamma_0(D)\) with primitive character, of the Petersson-Kuznetsov summation formula and of the properties of Rankin-Selberg and symmetric square \(L\)-functions that are needed in order to estimate the size of Fourier coefficients of automorphic forms; in view of the lack of easily accessible and usable references for the statements needed here this is highly welcome.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11R29 Class numbers, class groups, discriminants

Citations:

Zbl 1163.11325
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