Power residues on Abelian varieties. (English) Zbl 1025.11019

Wong, Siman

Manuscr. Math. 102, No. 1, 129-137 (2000).

Let \(A\) be an Abelian variety defined over a number field \(k\), let \(P\in A(k)\) be a point and \(n>1\) an integer. One says that the \(n\)-power residue problem is true for \(P\), if the fact that the residue of \(P\) mod \({\mathfrak p}\) is an \(n\)th power in the reduction of \(A\) mod \({\mathfrak p}\) for almost all primes \({\mathfrak p}\) of \(k\), implies that \(P\) itself is an \(n\)th power in \(A\). Fix an algebraic closure \(\overline k\) of \(k\) and denote \(k_n=k(A[n])\). The following theorem is the main result of the paper:
The \(n\)th power residue problem is true for any nonzero point \(P\) of \(A(k)\) if either \(H^1(\text{Gal}(k_n/k)\), \(A[n])=0\) or \(n\) is a prime number and \(A\) is an elliptic curve.
This is related to a theorem of M. V. Nori [Invent. Math. 88, 257-275 (1978; Zbl 0632.20030)] concerning the vanishing of \(H^1(\text{Gal} (k_l/k)\), \(A[l])\) for primes \(l\). Similar statements are proved for a finite set of points of \(A(k)\).

Reviewer: Herbert Lange (Erlangen)

MSC:

11G10	Abelian varieties of dimension \(> 1\)
11G05	Elliptic curves over global fields
20G40	Linear algebraic groups over finite fields

Keywords:

\(n\)-power residue problem; abelian variety

Citations:

Zbl 0632.20030