##
**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.

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

### MSC:

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.) |