Torsion points on curves and p-adic abelian integrals. (English) Zbl 0578.14038

For an elliptic curve C over a number field K, i.e. a curve of genus \(g=1\) having a rational point over K, the group \(T_ K\) of rational torsion points on C over K is finite by the Mordell-Weil theorem, and the (still unproved) boundedness conjecture implies that \(T_ K\) is bounded by a constant depending only on K. For (smooth) curves C of higher genus \(g\geq 2\) in an abelian variety J, the generalized Manin-Mumford conjecture states the finiteness of the set \(T_{\bar K}\) of torsion points of J arizing from C and being defined over the algebraic closure \(\bar K\) of K. This latter conjecture was proved by S. Lang for abelian varieties J admitting complex multiplication (CM) and by Raynaud for arbitrary abelian varieties. Assuming that the abelian variety J has potential complex multiplication, the author proves the following boundedness theorem: \(\#T_{\bar K}\leq pg,\) where p denotes the smallest prime of \({\mathbb{Q}}\) divisible by a prime in the set of primes \({\mathfrak p}\) of the number field K such that (i) \({\mathfrak p}\) does not divide 2 or 3, (ii) K is unramified at \({\mathfrak p}\), (iii) C has good ordinary reduction over K at \({\mathfrak p}.\)
Generalizations of this theorem are also obtained. Moreover, the theorem can be applied to completely determining the torsion points on the Fermat curves \(F(m):X^ m+Y^ m+Z^ m=0\) provided that \(m+1\) is a prime and \(m\geq 10.\)
As a tool for proving his theorem, the author develops a theory of p-adic abelian integrals based on Tate’s rigid analysis and Monsky-Washnitzer’s dagger analysis. In particular, the p-adic integrals of the first kind on an abelian variety turn out to satisfy an addition law as a result of which the torsion points on a curve C in its Jacobian J can be identified as the common zeros of these integrals. This establishes the connection between torsion points and p-adic abelian integrals.
Some interesting examples of torsion points on genus \(g=2\) curves are given at the end of the paper showing, among other things, that the primes 2 and 3 play a special role and that the CM-hypothesis is indispensable in the theorem.
Reviewer: H.G.Zimmer


14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14G20 Local ground fields in algebraic geometry
14K22 Complex multiplication and abelian varieties
14G05 Rational points
14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
14H05 Algebraic functions and function fields in algebraic geometry
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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