Galois theory and torsion points on curves.(English)Zbl 1065.11045

Let $$K$$ be a number field, $$\overline K$$ an algebraic closure for $$K$$ and $$X$$ an algebraic curve over $$K$$ of genus $$g\geq 2$$. Assume that $$X$$ is embedded in its Jacobian $$J$$ via a $$K$$-rational Albanese map $$i: X\to J$$. The Manin-Mumford conjecture states that the set $$X(\overline K)\cap J(\overline K)^{\text{tors}}$$ is finite.
The first proof of this conjecture was provided by M. Raynaud [Invent. Math. 74, 207–233 (1983; Zbl 0564.14020) and some years later a second by R. F. Coleman [Duke Math J. 541, 615–640 (1987; Zbl 0626.14022)]. In case where $$X= X_0(p)$$, with $$p$$ a prime $$\geq 23$$, the Coleman-Kaskel-Ribet conjecture states that the set of torsion points on $$X$$ in the embedding $$i_\infty:X\to J$$ is precisely $$\{0,\infty\}\cup H$$, where $$H$$ is the set of hyperelliptic branch points on $$X$$ when $$X$$ is hyperelliptic and $$p\neq 37$$ and $$H=\varnothing$$ otherwise.
In the paper under review, the authors deal with Galois-theoretic techniques for studying torsion points on curves and give new proofs of the above results.

MSC:

 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11G18 Arithmetic aspects of modular and Shimura varieties 14H25 Arithmetic ground fields for curves

Citations:

Zbl 0564.14020; Zbl 0626.14022
Full Text:

References:

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