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The endomorphism rings of jacobians of cyclic covers of the projective line. (English) Zbl 1058.14064

The author proves the following theorem.
Theorem. Suppose that \(K\) is a field of characteristic zero, \(K_a\) its algebraic closure, and that \(f(x) \in K[x]\) is an irreducible polynomial of degree \(n \geq 5\), whose Galois group coincides either with the full symmetric group \(S_n\) or with the alternating group \(A_n\). Let \(p\) be an odd prime, \(\mathbb{Z}[\zeta_p]\) be the ring of integers in the \(p\)th cyclotomic field \(\mathbb{Q}(\zeta_p)\). Suppose that \(C\) is the smooth projective model of the affine curve \(y^p=f(x)\) and \(J(C)\) the Jacobian of \(C\). Then the ring \(\text{End}(J(C))\) of \(K_a\)-endomorphisms of \(J(C)\) is canonically isomorphic to \(\mathbb{Z}[\zeta_p]\) (note that \(\zeta_p\) represents the automorphism of \(J(C)\) induced by \((x,y) \mapsto (x,\zeta_p y)\)).
The proof relies on theorems of one of the author’s previous articles [J. Reine Angew. Math. 544, 91–110 (2002; Zbl 0988.14015)]. It was proved there that under the hypothesis of the theorem above, \(\mathbb{Q}[\zeta_p]\) is a maximal commutative subalgebra in \(\text{End}^0(J(C))\) which contains the center \(\mathcal{S}\). To finish the proof the author shows that if \(\mathcal{S} \neq \mathbb{Q}[\zeta_p]\), we would get a contradiction. This part is based on the following general result, proved at the beginning of the article. Let \(Z\) be a complex abelian variety and \(E/\mathbb{Q}\) be a Galois-subfield of \(\text{End}^0(Z)\) containing the center \(\mathcal{S}\). Let \(\Sigma_E\) be the set of embeddings of \(E \hookrightarrow \mathbb C\) and for \(\sigma \in \Sigma_E\) define \[ \text{Lie}(Z)_{\sigma}=\{x \in \text{Lie}(Z) | ex=\sigma(e) x \; \forall e \in E\}. \] Let \(n_{\sigma}=\dim_{\mathbb C} \text{Lie}(Z)_{\sigma}\). Then if \(\mathcal{S} \neq E\) there exists a nontrivial automorphism \(\kappa : E \rightarrow E\) such that \(n_{\sigma}=n_{\sigma \kappa}\) for all \(\sigma \in \Sigma_E\).

MSC:

14K15 Arithmetic ground fields for abelian varieties
14H40 Jacobians, Prym varieties
11G10 Abelian varieties of dimension \(> 1\)

Citations:

Zbl 0988.14015
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