A $p$-adic algorithm to compute the Hilbert class polynomial

A p-adic algorithm to compute the Hilbert class polynomial

Classicaly, the Hilbert class polynomial P∆ ∈ Z[X] of an imaginary quadratic discriminant ∆ is computed using complex analytic techniques. In 2002, Couveignes and Henocq [5] suggested a p-adic algorithm to compute P∆. Unlike the complex analytic method, it does not suffer from problems caused by rounding errors. In this paper we complete the outline given in [5] and we prove that, if the Genera...

22.1 the Hilbert Class Polynomial

The appellation “Hilbert” is sometimes reserved for cases where D is a fundamental discriminant (in which case HD(X) is more generally called a ring class polynomial), but we shall use the term Hilbert class polynomial to refer to HD(X) in general. Our first objective is to use the fact that ΦN ∈ Z[X,Y ] to prove that HD ∈ Z[X]. We require the following lemma. Lemma 22.2. If N is prime then the...

p-ADIC CLASS INVARIANTS

We develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.

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We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of the affine space A over a p-adic field, endowed with polynomial actions on each coordinate of A. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

The Probability That a Random Monic p-adic Polynomial Splits