# Six Short Chapters on Automorphic Forms and L-functions

## Ze-Li Dou

Language: English

Pages: 123

ISBN: 3642287077

Format: PDF / Kindle (mobi) / ePub

"Six Short Chapters on Automorphic Forms and L-functions" treats the period conjectures of Shimura and the moment conjecture. These conjectures are of central importance in contemporary number theory, but have hitherto remained little discussed in expository form. The book is divided into six short and relatively independent chapters, each with its own theme, and presents a motivated and lively account of the main topics, providing professionals an overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevant literature.

Ze-Li Dou and Qiao Zhang are both associate professors of Mathematics at Texas Christian University, USA.

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purely imaginary numbers such that s1 + s2 = 0. Let ε1 , ε2 be two elements in the set {0, 1}. Then it is obvious that the mapping χi : R× → C given by x → sgn(x)εi |x|si is a character for each i = 1, 2. Thus so is χ∞ : o × → C deﬁned by χ∞ (a) = χ1 (a)χ2 (a). We now choose a speciﬁc χ∞ so that the product χ = χ∞ χm is trivial on o× . Once this strategy is decided upon, the rest is carried out without conceptual diﬃculty. Consider ﬁrst the group of totally positive units, × o× + ⊂ o . 46

that μ k 2 = 0, i.e., L k ,f 2 N k 2 (|t| + 2)2 ε . The same problem can be formulated for general automorphic L-functions and, in particular, the classical Riemann zeta function. In many arithmetic problems, especially in equidistribution problems, quite often all we need is simply a subconvexity bound, and the speciﬁc values of the order function μ(k/2) are inconsequential to our conclusions. As 88 Special Values of L-functions Chapter 5 an example, we conclude this section by

R-linear isomorphisms Ba ∼ = M2 (R)δ × Hδ (6.2) Automorphic forms with respect to BE 6.2 101 and (BE )a ∼ = M2 (R)ζ × Hζ . (6.3) Here H denotes the ring of Hamilton quaternions, and the subscript a indicates the inﬁnite part of the adelized space under consideration. The adelization itself will be denoted by the subscript A, and its ﬁnite part will be given the subscript h. Thus, for example, BA = Ba Bh . For each v ∈ a, we ﬁx an extension u ∈ aE once and for all. The collection of these

r; h) 117 In fact, the mapping f → fA is an injection of Sk (Γ, φ, λ) into the space of all functions g on GA satisfying equation (6.35) with fA replaced by g. Suppose f ∈ Sk (c, Ψ), and consider the form f1 ∈ Sk (Γ, φ, λ) as in (6.31). The restriction of f to GA coincides with (f1 )A as deﬁned by (6.34): f |GA = (f1 )A . Let y ∈ GA ∩ FA× Y . Then we can ﬁnd a ﬁnite subset W ⊂ Gh such Dn w. Dn w and Dn yDn = that we simultaneously have Dn y Dn = w∈W w∈W Furthermore, there exists an element

we have considered in the previous sections are deﬁned in the same setting; they all involve the automorphic forms in a more or less direct manner. It is therefore natural to ask whether these periods are related, and, if they are, how they are related to another. We also observe that, so far, we have restricted ourselves to the simplest possible case of F = Q. Can the notion of the various periods of automorphic forms be generalized to more general types of automorphic forms? Shimura’s period