By William Stein
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Extra resources for Algorithms for Computing with Modular Forms
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 .