Download Algorithms for Computing with Modular Forms by William Stein PDF

By William Stein

Show description

Read or Download Algorithms for Computing with Modular Forms PDF

Best applied mathematicsematics books

The Political Power of Business: Structure and Information in Public Policymaking (Toutledge Research in Comparative Politics)

This e-book analyzes the impression of industrial in democratic politics. recommendation from enterprise actors on a regular basis contains extra weight with policymakers than different pursuits since it refers back to the center of the state-market nexus in democratic capitalism: the implications for citizens and policymakers of harming company and the financial system.

Company Law (Principles Of Law)

Corporation legislations is a development region extra so, most likely, than the other sector in legislations. It reaches out into different parts of legislations and, after all, new parts of legislation are continuously rising, for instance auditors negligence, funding legislations and the FSA and management orders. The ebook supplies an entire research of the subsequent parts: the corporate and different company businesses; varieties of corporation; developing the corporate; dealing with the corporate; reconstituting the corporate; and supervision of corporation legislations.

The Linearization Method in Hydrodynamical Stability Theory

This e-book offers the speculation of the linearization process as utilized to the matter of steady-state and periodic motions of constant media. the writer proves infinite-dimensional analogues of Lyapunov's theorems on balance, instability, and conditional balance for a wide type of continuing media.

Extra resources for Algorithms for Computing with Modular Forms

Example text

That if g ∈ (Z/pr Z)∗ and g n/pi = 1 for all pi | n = ϕ(n), then g is a generator of (Z/pr Z)∗ . 3 Let p be an odd prime and n ≥ 2 an integer, and prove that (1 + pn−1 (Z/pn Z), ×) ∼ = (Z/pZ, +). Use this to show that solving the discrete log problem in (Z/pn Z)∗ is “not much harder” than solving the discrete log problem in (Z/pZ)∗ . 36 CHAPTER 2. 4 Suppose ε is a nontrivial Dirichlet character modulo 2n of order r over the complex numbers C. Prove that the conductor of ε is c= 2ord2 (r)+1 2ord2 (r)+2 if ε(5) = 1 if ε(5) = 1.

Bn ] of integers, but this time with bi ∈ Z/ gcd(si , n)Z, where si is the order of gi . Then ε(gi ) = ζ bi ·n/(gcd(si ,n)) , which is already complicated enough to ring warning bells. 4. EXERCISES 35 tation we set up an identification D(N, R) ∼ = Z/ gcd(si , n)Z, i and arithmetic is efficient. This approach is seductive because every sequence of integers determines a character, and the sizes of the integers in the sequence nicely indicate the local orders of the character. However, giving analogues of many of the algorithms discussed in this chapter that operate on characters represented this way is tricky.

6 (Conductor). This algorithm computes the conductor of a Dirichlet character ε ∈ D(N, R). 1. 3, find characters εi whose product is ε. 2. 1, compute the orders ri of each εi . 3. , of order 1), or set ci ← pi i of p that divides n. 4. ] If p1 = 2 and ε1 (5) = 1, set c1 ← 2c1 . 5. [Finished] Output c = ci and terminate. 32 CHAPTER 2. DIRICHLET CHARACTERS Proof. Let εi be the local factors of ε, as in Step 1. We first show that the product of the conductors fi of the εi is the conductor f of ε. Since εi factors through (Z/fi Z)∗ , the product ε of the εi factors through (Z/ fi Z)∗ , so the conductor of ε divides fi .

Download PDF sample

Rated 4.17 of 5 – based on 46 votes