O'Reilly logo

Convolution and Equidistribution by Nicholas M. Katz

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 28

The Situation over Z: Results

Suppose we are given an integer monic polynomial f (x) Z[x] of degree n ≥ 2 which, over C, is “weakly supermorse,” meaning that it has n distinct roots in C, its derivative f ′ (x) has n – 1 distinct roots (the critical points) αi C, and the n – 1 values f (αi) (the critical values) are all distinct in C. Denote by S the set of critical values. Suppose that S is not equal to any nontrivial multiplicative translate aS, for any a ≠ 1 in C×. It is standard that for all but finitely many primes p, the reduction mod p of f will satisfy all the hypotheses of Theorem 17.6. Let us say such a prime p is good for f.

Choose a prime, ℓ and a field isomorphism ι : Q C. For each p ≠ which is good for f, form ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required