A New Class of $$p$$-Adic Lipschitz Functions and Multidimensional Hensel’s Lemma

Extension of P-adic Definable Lipschitz Functions

Write OK for the valuation ting, MK for the maximal ideal of K and kK for the residue field. Let us fix $ some uniformizer of K. We denote by acm : K → OK/(MK) the map sending some nonzero x ∈ K to x$−ord(x) mod MK , and sending zero to zero. This is a definable map. We denote by RV the union of K×/(1 +MK) and {0} and by rv : K → RV the quotient map. More generally, if m ∈ N∗, we set RVm = K×/(...

Varianten von Hensels Lemma

Hensel’s lemma, valuations, and p-adic numbers

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.

p-ADIC PERIODS, p-ADIC L-FUNCTIONS, AND THE p-ADIC UNIFORMIZATION OF SHIMURA CURVES