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Explicit descent for Jacobians of cyclic covers of the projective line. (English) Zbl 0888.11023

Let \(p\) be a prime, and let \(k\) be a global field not of characteristic \(p\) and containing a primitive \(p\)th root of unity \(\zeta\). Consider a curve \(X\) defined over \(k\) by an equation of the form \(y^p =f(x)\) (where \(f\) has no zeros of multiplicity \(p\) or greater), and let \(J\) denote its Jacobian. The curve \(X\) obviously has an automorphism of order \(p\) given by multiplying the \(y\)-coordinate by \(\zeta\). This induces an automorphism on \(J\), which by abuse of notation will also be denoted \(\zeta\). Then \(\phi = 1 - \zeta\) is an isogeny \(J \longrightarrow J\) of fairly small degree. The paper under review deals with the problem of determining (the size of) the corresponding Selmer group \(Sel^\phi(J,k)\) (which then gives a bound on the Mordell-Weil rank of \(J(k)\)).
In an earlier paper [Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann., to appear], E. F. Schaefer has dealt with the special case that \(\deg f\) is prime to \(p\) (and \(f\) is square-free). This essentially means that the \(x\)-coordinate map \(X \longrightarrow \mathbb{P}^1\) has a \(k\)-rational ramification point. He identifies \(H^1(k, J[\phi])\) with a subgroup of \(L^\times/(L^\times)^2\), where \(L = k[T]/f(T)\), and gives an explicit description of the coboundary map from \(J(k)\) to \(L^\times/(L^\times)^2\). The same holds for all completions of \(k\). Together with some information on the behaviour at good primes, this yields a practical algorithm for computing the Selmer group.
The paper under review tries to generalize this approach to the case where \(\deg f\) is a multiple of \(p\) (which is the general case) and where \(f\) is not necessarily square-free. Let \(f_0\) be the radical of \(f\) (i.e.the product of all the irreducible polynomials dividing \(f\)), and let \(L = k[T]/f_0(T)\). The authors define an explicit map called ‘\(x-T\)’ from the subgroup of \(J(k)/\phi J(k)\) represented by \(k\)-rational divisors into \(L^\times/((L^\times)^2 k^\times)\). This map is not as directly related to the coboundary map as in the special case discussed in the preceding paragraph, but it is very close to it. The paper gives a detailed explanation of the precise relationship. The consequences for the Selmer group are as follows. Suppose that \(X\) has a \(k_v\)-rational divisor class of degree one for every place \(v\) of \(k\) (then all of \(J(k)/\phi J(k)\) is represented by \(k\)-rational divisors). We can use the \(x-T\) map (for \(k\) and its completions) to compute a ‘fake Selmer group’. It is a homomorphic image of the usual Selmer group by a map whose kernel is either trivial or of order \(p\). The authors give a simple criterion for deciding which of the two is the case.
The methods developed in this paper make it possible to find, or at least bound, the Mordell-Weil rank for quite general curves of the type given at the beginning of this review. The authors demonstrate this by including as a worked example the determination of the Mordell-Weil rank of a non-hyperelliptic curve of genus \(8\) over \(\mathbb{Q}(\sqrt{-3})\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H40 Jacobians, Prym varieties
11G10 Abelian varieties of dimension \(> 1\)
14H25 Arithmetic ground fields for curves
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