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Families of absolutely simple hyperelliptic Jacobians. (English) Zbl 1186.14031

Let \(K\) be a field of characteristic different from \(2\), and let \(f(x)\) be a polynomial of degree \(n\geq 5\) in \(K[x]\) without multiple roots. Let \(X/K\) be the Jacobian variety of the hyperelliptic curve \(y^2=f(x)\), and let \(\text{End}(X)\) denote the ring of endomorphisms on \(X\) defined over the algebraic closure of \(K\). In the author’s earlier works, it is proved that if the action of \(\text{Gal}(K)\) on the roots of \(f(x)\) is the symmetric group \(S_n\) or the alternating group \(A_n\), then \(\text{End}(X) =\mathbb Z\), provided that \(\text{Char}(K)\neq 3\) or \(n\geq 7\).
In the article under review, the author proves similar results for \(f(x)=(x-t)h(x)\) where \(t\in K\) and the action of \(\text{Gal}(K)\) on the roots of \(h(x)\) is \(S_{n-1}\) or \(A_{n-1}\). If \(n\) is even, under a change of variables the problem is reduced to the author’s earlier work, but the case of \(n\) being odd turns out to be more difficult. From now on, let \(f(x)\) be \((x-t)h(x)\) as described above, and let \(\text{End}^0(X):=\text{End}(X) \otimes \mathbb Q\). The author proves that if \(n \geq 9\) is odd, then one of the following conditions holds: (i) \(\text{End}^0(X)\) is \(\mathbb Q\) or a quadratic extension of \(\mathbb Q\); (ii) \(\text{Char}(K)>0\) and \(X\) is supersingular; furthermore, if \(\text{Char}(K)=0\) and \(n\geq 11\), then \(\text{End}(X)=\mathbb Z\). In [J. S. Ellenberg et al., J. Lond. Math. Soc., II. Ser. 80, No. 1, 135–154 (2009; Zbl 1263.11064)], a similar result is proved for \(K\) being finitely generated over \(\mathbb Q\): if \(h(x) \in K[x]\) is an arbitrary polynomial of positive even degree without multiple roots, then for all but finitely many \(t \in K\) the Jacobian variety \(X\) is absolutely simple. The approaches in this work is based on arithmetic geometry and analytic number theory while the approach in this article under review is purely algebraic. In the later part of the paper the author establishes analogous results in the context of \(\ell\)-adic Lie groups and their Lie algebras, and prove for the Jacobian variety \(X\) the Tate and Hodge conjectures as well as the Mumford-Tate conjecture.
Let us conclude this review by sketching the outline of the proofs of the main results stated above when the action of \(\text{Gal}(K)\) on the roots of \(h(x)\) is \(A_{n-1}\). In fact, the key lemmas are established in his earlier work [A. Elkin and Y. G. Zarhin, J. Ramanujan Math. Soc. 21, No. 2, 169–187 (2006; Zbl 1115.14024); Math. Ann. 340, No. 2, 407–435 (2008; Zbl 1222.14084)], and in this paper the author proves that the conditions for those lemmas to be applied are always satisfied. It was proved in [J. Pure Appl. Algebra 77, 253–262 (1992; Zbl 0808.14037)] that all endomorphisms on an abelian varieties are defined over the \(4\)-division field. Let \(\text{End}^0(X):=\text{End}(X)\otimes \mathbb Q\). Let \(\widetilde{G}_4\) be the action of \(\text{Gal}(K)\) on the \(4\)-division field of \(X\), so that there is a map \(\widetilde{G}_4 \to \text{Aut}( \text{End}^0(X) )\), and let \(\widetilde{G}_2\) be the action on the \(2\)-division field. Let \(\text{End}_K^0(X)\) be the \(\mathbb Q\)-subalgebra generated by endomorphisms defined over \(K\), so that \(\text{End}_K^0(X) = \text{End}^0(X)^{\widetilde{G}_4}\), and let \(C\) be the center of \(\text{End}^0(X)\). Let \(X_2\) denote the group of \(2\)-torsion points.
In the author’s earlier work the following lemmas are proved: Lemma I: If \(X_2\) does not contain a proper non-trivial \(\widetilde{G}_2\)-(setwise) invariant even dimensional subspace and the centralizer \(\text{End}_{\widetilde{G}_2}(X_2)\) has \(\mathbb F_2\)-dimension \(2\), then \(\text{End}_K^0(X)\) is either \(\mathbb Q\) or a quadratic extension of \(\mathbb Q\);
Lemma II: Let \(g:=\dim(X)\), and assume that \(C\) is a field. Then, (1) \(\dim_\mathbb Q( \text{End}^0(X) )\) divides \(4g^2\), and (2) if \(\dim_\mathbb Q(\text{End}^0(X))=4g^2\), then \(\text{Char}(K)>0\) and \(X\) is supersingular.
In the paper under review he proves in a purely abstract setting that the conditions in Lemma I are satisfied, using the assumption that the action of \(\text{Gal}(K)\) on the roots of \(h(x)\) is the alternating group \(A_{2g}\) where \(g=\dim(X)\). Thus, \(\text{End}^0_K(X)\) is \(\mathbb Q\) or a quadratic extension of \(\mathbb Q\), and the remaining case is \(\text{End}_K^0(X) \neq \text{End}^0(X)\), for which the author proves, as sketched below, that Lemma II, (2) must be the case. Let us sketch the proof of the fact that \(C\) is \(\mathbb Q\) or a quadratic extension of \(\mathbb Q\); in particular, it is a field as required in Lemma II. Without loss of generality, he argues that we may assume that the surjective map \(\widetilde{G}_4 \to \widetilde{G}_2\) is a minimal cover, from which he deduces a strong restriction on \(\widetilde{G}_4\) assuming that \(\widetilde{G}_2\cong A_{2g}\): the group \(\widetilde{G}_4\) does not contain a proper subgroup of index less than \(2g\). If \(C = \bigoplus_1^r C_i\) is a decomposition of the semi-simple algebra \(C\) into fields \(C_i\)’s, then \(1\leq r \leq g=\dim(X)\), and the restriction on the index of subgroups implies that \(\widetilde{G}_4\) acts trivially on the set \(\{ C_1,\dots,C_r\}\). The trivial action on this set implies that the subalgebra \( \bigoplus_1^r \mathbb Q\) of \(C\) is elementwise fixed by \(\widetilde{G}_4\), and hence, the subalgebra is contained in \(\text{End}^0(X)^{\widetilde{G}_4}=\text{End}_K^0(X)\) which is a field (with no zero divisors). This proves that \(r=1\), and hence, \(C\) is a field. The author also uses the restriction on index and classic results on the ring of endomorphisms on an abelian variety to further prove that every element of \(C\) is in fact fixed by \(\widetilde{G}_4\), so \(C\subset \text{End}_K^0(X)\), which proves the assertion about \(C\). What remains to finish the proof of his main result is to show that \(\dim_\mathbb Q(\text{End}^0(X))=4g^2\) must be always the case. Let \(C_K\) be the centralizer of \(\text{End}_K^0(X)\) in \(\text{End}^0(X)\). Since it is proved in this paper that \(C\) sits inside \(\text{End}_K^0(X)\), he is able to use from his earlier work a formula for \(d=\dim_{\text{End}_K^0(X)}(C_K)\) and a result about the existence of a minimal cover of \(\widetilde{G}_2\) by a subgroup of the unit group \(C_K^*\), and pulls out a contradiction on the size \(d\) if \(d\neq 4g^2\), using the assumption \(\widetilde{G}_2\cong A_{2g}\).

MSC:

14H40 Jacobians, Prym varieties
14K05 Algebraic theory of abelian varieties
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G10 Abelian varieties of dimension \(> 1\)
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