Geometry of p-jets

The basic geometry of Witt vectors. II: Spaces

“…The construction of the full functor had to wait until Joyal [32] and, independently, Buium [9]. It also has applications in number theory, most notably in the study of p-adic points on varieties.…”

“…It also has applications in number theory, most notably in the study of p-adic points on varieties. For example, see Buium [9] and Buium-Poonen [11] (as well as Buium's earlier work [8] for applications of analogous constructions in complex algebraic geometry). These two adjoint constructions see different sides of the arithmetic of A-the ring W n (A) sees certain maps into A, and the ring Λ n ⊙ A sees certain maps out of it.…”

Plane curves and p-adic roots of unity

“…We write 771 -1 + •K and notice that the inequality |/(Ci,C2)| < P~l implies / ( l + TT, (1 + 7r) r ) = O^odTr"-1 ). On the other hand if O is the ring of integers of the field F(r)i), where F is a unramified extension of Q p containing the coefficients of / , then O/n p~1 is isomorphic to k[t)/t p~l , where k is the residue field of F. Therefore we obtain / 0 ( l + 1 , (1 + t) r ) = 0(modt p - 1 ).…”

“…First of all, if / is a polynomial in just one variable and is irreducible, then it is a binomial and we are done. Therefore, we may assume that there exists i\,ji with aoj 1 ,<ii 1 …”

“…U REMARKS, (i) If X is a projective curve of genus bigger than one embedded in an Abelian variety A, all defined over an unramified extension of Q p , then Raynaud [4] proved that there are only finitely many torsion points of A of order prime to p which are in X modulo p 2 and Buium [1] gave an explicit bound for the number of those points. Perhaps the techniques of Coleman [2] could be used to extend this result to the full torsion and obtain an Abelian analogue of the above result.…”