By Goro Shimura
A ebook on any mathematical topic past the textbook point is of little price except it comprises new rules and new views. It is helping to incorporate new effects, only if they offer the reader new insights and are awarded besides identified outdated ends up in a transparent exposition. it truly is with this philosophy that the writer writes this quantity. the 2 topics, Dirichlet sequence and modular varieties, are conventional topics, yet right here they're handled in either orthodox and unorthodox methods. whatever the unorthodox therapy, the writer has made the booklet available to people who usually are not conversant in such themes through together with lots of expository fabric.
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Extra info for Elementary Dirichlet Series and Modular Forms (Springer Monographs in Mathematics)
Example text
Then |β | = 1 for every σ ∈ Gal(K/Q). By Kronecker’s theorem, β ∈ W. Suppose m is odd. Then β = εγ 2 with γ ∈ W and ε = ±1. If ε = −1, we have (αγ)ρ = −αγ, and so 2αγ = αγ − (αγ)ρ ∈ d(K/F ), where d(K/F ) is the different of K relative to F. 12a)) and prime to 2. Thus ε = 1, and so (αγ)ρ = αγ, which shows that αγ ∈ F. Therefore α ∈ W rF as expected. Suppose m = 2r ; let n = 2r−1 and β = ζ −a with a ∈ Z. Then ζ n = −1. If a is even, we can put ζ = γ 2 with γ ∈ W, which together with the 42 II. CRITICAL VALUES OF DIRICHLET L-FUNCTIONS above argument leads to the desired conclusion; so assume a to be odd.
5) rather easily. We present the first type as eight formulas depending on the nature of the Dirichlet character and the parameter c of the generalized Euler polynomial Ec,n . 2), is somewhat different from the other seven. Though it may be possible to state those seven as a single formula, we do not do so, as such would make it cumbersome and less easy to understand. In the following theorem, the product χλ for two characters χ and λ means the character defined by (χλ)(m) = χ(m)λ(m), which can be imprimitive if the conductors of χ and λ are not relatively prime.
The formula is valid even for d = 1 and trivial χ, in which case G(χ) = 1 and the sum on the left-hand side is Ec,k−1 (1/2). (vii) Suppose d is prime to 3; let λ be the Dirichlet character modulo 9 such that λ(2) = e(2/3). e(ε/9){1 + e(1/6)} j=1−g −k k · (2πi) 3 χ(3)G(χ) · L(k, χλε ) if χ(−1) = (−1)k , L(k, µ3 χλε ) if χ(−1) = (−1)k+1 , where c = −e(1/3), and g = [d/3] + 1 or g = −[d/3] according as ε = 1 or ε = −1. The formula is valid even for d = 1 and trivial χ, in which case G(χ) = 1 and the sum on the left-hand side is Ec,k−1 (1/3) if ε = 1 and Ec,k−1 (2/3) if ε = −1.