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The Cassels-Tate pairing on polarized Abelian varieties. (English) Zbl 1024.11040
Authors’ introduction: The study of the Shafarevich-Tate group \(\text{ Ш}(A)\) of an Abelian variety \(A\) over a global field \(k\) is fundamental to the understanding of the arithmetic of \(A\). It plays a role analogous to that of the class group in the theory of the multiplicative group over an order in \(k\). J. W. S. Cassels [J. Reine Angew. Math. 211, 95-112 (1962; Zbl 0106.03706)] in one of the first papers devoted to the study of \(\text{ Ш}\), proved that in the case where \(E\) is an elliptic curve over a number field, there exists a pairing \[ \text{ Ш}(E)\times\text{ Ш}(E)\to\mathbb{Q}/ \mathbb{Z} \] that becomes nondegenerate after one divides \(\text{ Ш}(E)\) by its maximal divisible subgroup. He proved also that this pairing is alternating; i.e., that \(\langle x,x\rangle=0\) for all \(x\). If, as is conjectured, \(\text{ Ш}(E)\) is always finite, then this would force its order to be a perfect square. J. Tate [Proc. Int. Cong. Math., Stockholm, 1962, 288-295 (1963; Zbl 0126.07002)] soon generalized Cassels’ results by proving that for Abelian varieties \(A\) and their duals \(A^\vee\) in general, there is a pairing \[ \text{ Ш}(A)\times \text{ Ш}(A^\vee)\to \mathbb{Q}/ \mathbb{Z} \] that is nondegenerate after division by maximal divisible subgroups. He also proved that if \(\text{ Ш}(A)\) is mapped to \(\text{ Ш}(A^\vee)\) via a polarization arising from a \(k\)-rational divisor on \(A\) then the induced pairing on \(\text{ Ш}(A)\) is alternating. But it is known that when \(\dim A>1\), a \(k\)-rational polarization need not come from a \(k\)-rational divisor on \(A\). (See Section 4 for the obstruction.) For principally polarized Abelian varieties in general M. Flach [J. Reine Angew. Math. 412, 113-127 (1990; Zbl 0711.14001)] proved that the pairing is antisymmetric, by which we mean \(\langle x,y\rangle= -\langle y,x\rangle\) for all \(x,y\), which is slightly weaker than the alternating condition.
It seems to have been largely forgotten that the alternating property was never proved in general: in a few places in the literature, one can find the claim that the pairing is always alternating for Jacobians of curves over number fields, for example. In Section 10 we will give explicit examples to show that this is not true, and that \(\#\text{ Ш}(J)\) need not be a perfect square even if \(J\) is a Jacobian of a curve over \(\mathbb{Q}^2\).
One may ask what properties beyond antisymmetry the pairing has in the general case of a principally polarized Abelian variety \((A,\lambda)\) over a global field \(k\). For simplicity, let us assume here that \(\text{ Ш}(A)\) is finite, so that the pairing is nondegenerate. Flach’s result implies that \(x\mapsto\langle x,x\rangle\) is a homomorphism \(\text{ Ш}(A)\to\mathbb{Q}/ \mathbb{Z}\), so by nondegeneracy there exists \(c\in\text{ Ш}(A)\) such that \(\langle x,x\rangle= \langle x,c\rangle\). Since Flach’s result implies \(2\langle x,x\rangle=0\), we also have \(2c=0\) by nondegeneracy. It is then natural to ask, what is this element \(\text{ Ш}(A)[2]\) that we have canonically associated to \((A,\lambda)\)? An intrinsic definition of \(c\) is given in Section 4, and it will be shown that \(c\) vanishes (i.e., the pairing is alternating) if and only if the polarization arises from a \(k\)-rational divisor on \(A\). This shows that Tate’s and Flach’s results are each best possible in a certain sense.
Our paper begins with a summary of most of the notation and terminology that will be needed, and with two definitions of the pairing. (We, give two more definitions and prove the compatibility of all four in an appendix.) Sections 4 and 5 give the intrinsic definition of \(c\) and show that it has the desired property. (Actually, we work a little more generally: \(\lambda\) is not assumed to be principal and in fact it may be a difference of polarizations.) Section 6 develops some consequences of the existence of \(c\); for instance if \(A\) is principally polarized and \(\text{ Ш}(A)\) is finite, then its order is a square or twice a square according as \(\langle c,c\rangle\) equals 0 or \({1\over 2}\) in \(\mathbb{Q}/\mathbb{Z}\). We call \(A\) even in the first case and odd in the second case.
The main goal of Sections 7 and 8 is to translate this into a more down-to-earth criterion for the Jacobian of a genus \(g\) curve \(X\) over \(k\): \(\langle c,c\rangle=N/2 \in\mathbb{Q}/ \mathbb{Z}\) where \(N\) is the number of places \(v\) of \(k\) for which \(X\) has no \(k_v\)-rational divisor of degree \(g-1\). Section 9 applies this criterion to hyperelliptic curves of even genus \(g\) over \(\mathbb{Q}\), and shows that a positive proportion \(\rho_g\) of these (in a sense to be made precise) have odd Jacobian. It also gives an exact formula for \(\rho_g\) in terms of certain local densities, and it determines the behavior of \(\rho_g\) as \(g\) goes to infinity. The result relating the local and global densities is quite general and can be applied to other similar questions. Numerical calculations based on the estimates and formulas obtained give an approximate value of \(13\%\) for the density \(\rho_2\) of curves of genus 2 over \(\mathbb{Q}\) with odd Jacobian.
Section 10 applies the criterion to prove that Jacobians of certain Shimura curves are always even. It gives also a few other examples, including an explicit genus 2 curve over \(\mathbb{Q}\) for whose Jacobian we can prove unconditionally that \(\langle c,c\rangle={1\over 2}\) and \(\text{ Ш}\cong \mathbb{Z}/2\mathbb{Z}\), and another for which \(\text{ Ш}\) is finite of square order, but with \(\langle , \rangle\) not alternating on it.
Finally, Section 11 addresses the analogous questions for Brauer groups of surfaces over finite fields, recasting an old question of Tate in new terms.

MSC:
11G10 Abelian varieties of dimension \(> 1\)
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H40 Jacobians, Prym varieties
14K15 Arithmetic ground fields for abelian varieties
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