Download Automorphic Forms, Representations, and L-functions by Armand Borel PDF

By Armand Borel

Comprises sections on Reductive teams, representations, Automorphic kinds and representations.

Show description

Read or Download Automorphic Forms, Representations, and L-functions PDF

Best number theory books

An Introduction to the Theory of Numbers

The 5th variation of 1 of the traditional works on quantity idea, written via internationally-recognized mathematicians. Chapters are rather self-contained for higher flexibility. New good points comprise improved remedy of the binomial theorem, thoughts of numerical calculation and a bit on public key cryptography.

Reciprocity Laws: From Euler to Eisenstein

This e-book is set the improvement of reciprocity legislation, ranging from conjectures of Euler and discussing the contributions of Legendre, Gauss, Dirichlet, Jacobi, and Eisenstein. Readers a professional in simple algebraic quantity conception and Galois thought will locate specific discussions of the reciprocity legislation for quadratic, cubic, quartic, sextic and octic residues, rational reciprocity legislation, and Eisenstein's reciprocity legislation.

Discriminant Equations in Diophantine Number Theory

Discriminant equations are a tremendous classification of Diophantine equations with shut ties to algebraic quantity conception, Diophantine approximation and Diophantine geometry. This e-book is the 1st finished account of discriminant equations and their functions. It brings jointly many points, together with powerful effects over quantity fields, powerful effects over finitely generated domain names, estimates at the variety of suggestions, functions to algebraic integers of given discriminant, strength crucial bases, canonical quantity platforms, root separation of polynomials and relief of hyperelliptic curves.

Extra resources for Automorphic Forms, Representations, and L-functions

Example text

0 s−k−2 0 0 . . 0 −|x0 + 2m| In the cases where j = 0 it simplifies to a diagonal matrix of the form −diag(|xj + 2m|−µ , . . , |xj + 2m|−µ , −(k + 1 − s)|xj + 2m|−µ , |xj + 2m|−µ , . . , |xj + 2m|−µ ) where the entry (k + 1 − s)(xj + 2m)−µ appears precisely at the j-th position. A summation of these matrices over the one-dimensional lattice 2Zej gives actually a well-defined invertible matrix for all z ∈ IRej \2Zej . Hence, all a-points of cot1,s,0 (z, 2Zej ) that lie on the xj axis are isolated.

15) which is s-monogenic in the variable a in 0 ≤ a < γ < ω . In particular, it is real analytic in B(0, γ) and can be thus represented in B(0, γ) by its Taylor series. Using the same argument as in the previous part of the proof, we obtain h(a) ≤ (|m|+|l|+k−s+1)! 11). (k + 1)k+3+|m|+|l|−s a ω k+|m|+|l|+s−2 . 1. ···lk ! After having applied the multinomial formula and an index shift we thus obtain g(z) ≤ ∞ |m|+k+2−s r=0 γ=2 ≤ L (r + γ) a + b ω k+|m|+3−s (k+1)R ω r (k + 1)k+3+|m|−s a + b ω k+|m|+3−s R R.

Xj + 2m|−µ , −(k + 1 − s)|xj + 2m|−µ , |xj + 2m|−µ , . . , |xj + 2m|−µ ) where the entry (k + 1 − s)(xj + 2m)−µ appears precisely at the j-th position. A summation of these matrices over the one-dimensional lattice 2Zej gives actually a well-defined invertible matrix for all z ∈ IRej \2Zej . Hence, all a-points of cot1,s,0 (z, 2Zej ) that lie on the xj axis are isolated. Since tan1,s,0 (z; 2Zej ) stems from cot1,s,0 (z, 2Zej ) by a applying a shift in the argument in the direction of the xj -axis, it follows directly that also the a-points of tan1,s,0 (z; 2Zej ) that lie on the xj axis are all isolated.

Download PDF sample

Rated 4.41 of 5 – based on 20 votes