Six Short Chapters on Automorphic Forms and L-functions
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"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.
use the same symbols for the W we have speciﬁed above. We choose the xλ such that their p-components are 1 for for every prime factor of c, and as usual xλ∞ = 1. This done, we deﬁne a character φλ on Γλ by φλ (γ) = ψY (x−1 λ γxλ ), ∀γ ∈ Γλ . The data for the deﬁnition of automorphic forms are ﬁnally all in place. We deﬁne h Mk,κ (c, ψ0 ) = Mk,κ (Γλ , φλ ), (2.14) Sk,κ (Γλ , φλ ). (2.15) λ=1 h Sk,κ (c, ψ0 ) = λ=1 These spaces are the same as the Mk,κ (W, Φ) and Sk,κ (W, Φ) in Section
the Dirichlet character χ must be a real quadratic character, that there can be at most one such exceptional zero, and that this exceptional zero, if exists, must be real. This conjecturally non-existing real zero is called the Siegel zero (associated to our choice of c). Much work has been done to eliminate the Siegel zero, or at least to push it as far to the left as possible. Let β˜ denote the mysterious Siegel zero associated to L(s, χ) with conductor N ; we would like to make it as small as
make it rigorous at the cost of technical complexity. Applying the two functional equations (5.13) and (5.14), we may extend the domain of Zm (s, w) to a much larger region, and the polar divisor w = 1 is also transformed to other polar divisors. Diaconu, Goldfeld and Hoﬀstein conjectured that Zm (s, w) can be analytically continued up to s = 1/2 and w = 1, and that all the polar divisors around the point (s, w) = (1/2, 1) are given by the images of w = 1. Then a careful analysis of the
looking form ∞ an q n . f (z) = n=0 1.3 Examples It is not surprising that the functions G2k (z) for k 2 all turn out to be modular forms. They are called Eisenstein series. The weight of G2k (z) is 2k. In fact the following identity is true: ∞ 2(2πi)2k σ2k−1 (n)q n . G2k (z) = 2ζ(k) + (2k − 1) n=1 (1.19) Here ζ is the Riemann zeta function, and the symbol σm is deﬁned by dm . σm (n) = d|n 6 Chapter 1 Modular forms and the Shimura-Taniyama Conjecture In general, if f and g are two
is a CM curve, then E and E are isogenous if and only if 22 Chapter 2 Periods of automorphic forms End(E) = End(E ). Moreover, if E ∼ = C/L and E ∼ = C/L , then the condition of isogeny is equivalent to the existence of some γ ∈ GL+ 2 (Q) such that γ(α ) = α, where α stands for the quotient of the two periods deﬁning L . Combining what we have thus recalled, we can deduce the following result. Let K be an imaginary quadratic ﬁeld. Then there exists a qK ∈ C, which depends only on K, such