5-torsion in the Shafarevich-Tate group of a family of elliptic curves
Let \(E\) be an elliptic curve over a number field \(k\). There is an abelian group attached to \(E\) known as the Shafarevich-Tate group of \(E/k\), and it is equipped with an alternating pairing known as the Cassels-Tate pairing. This pairing is important because if the Shafarevich-Tate group is finite (as it is conjectured to be), the pairing lets one decide whether principal homogeneous spaces of \(E\) are trivial or not, and the latter is needed to compute Mordell-Weil ranks via descent. Even without assuming the finiteness of the Shafarevich-Tate group, the pairing can be used to construct nontrivial elements of that group. The paper under review considers the following special case. Let \(E'\) be an elliptic curve over \(\mathbb Q\) whose \(5\)-torsion subgroup is isomorphic to \({\mathbb Z}/5 \oplus \mu_5\) as a Galois module. Let \(E\) be the quotient of \(E'\) by \({\mathbb Z}/5\), so that there is an isogeny \(\phi: E \rightarrow E'\) with kernel isomorphic to \(\mu_5\). The restriction of the Cassels-Tate pairing for \(E\) to the \(\phi\)-torsion in the Shafarevich-Tate group can be viewed as a pairing on the \(\phi\)-Selmer group \(S_\phi(E/{\mathbb Q})\). Descent theory implies that \(S_\phi(E/{\mathbb Q})\) can be identified with a finite subgroup of \({\mathbb Q}^*/{\mathbb Q}^{*5}\), and that this subgroup is effectively computable, at least in theory. The author first gives a very concrete description of the group \(S_\phi(E/{\mathbb Q})\) as a subgroup of \({\mathbb Q}^*/{\mathbb Q}^{*5}\) (Theorem 1.1), but her main result is that there is also a practical formula for the pairing on it in terms of local Hilbert norm residue symbols (Theorem 1.2). The precise statements and proofs of the theorems are expressed using the universal family of elliptic curves \(E'\) with \(5\)-torsion above, which was computed by \textit{K. Rubin} and \textit{A. Silverberg} [in Elliptic curves, modular forms, and Fermat's Last Theorem, 148-161 (1995; Zbl 0856.11027)]. To demonstrate the practicality of her formula, she uses it to show that the \(\phi\)-torsion of the Shafarevich-Tate group of the elliptic curve \[ E: y^2+y = x^3-x^2 -14128847434470x - 120901020992332992024 \] is isomorphic to \(({\mathbb Z}/5 {\mathbb Z})^5\). (This calculation is justified unconditionally by using the modularity of this elliptic curve, and the work of Kolyvagin and Gross and Zagier, to prove that \(E({\mathbb Q})\) is of rank 1.) Here we list a few related papers in the literature. \textit{J. W. S. Cassels} [J. Reine Angew. Math. 214/215, 65-70 (1964; Zbl 0236.14012)], \textit{R. Bölling} [Math. Nachr. 67, 157-179 (1975; Zbl 0314.14008)] and \textit{K. Kramer} [Proc. Am. Math. Soc. 89, 379-386 (1983; Zbl 0567.14018)] have given examples showing that the 2- and 3-torsion of Shafarevich-Tate groups can be arbitrarily large. \textit{W. McCallum} [Invent. Math. 93, 637-666 (1988; Zbl 0661.14033)] calculated the Cassels-Tate pairing to find nontrivial \(p\)-torsion in the Shafarevich-Tate group of the Jacobian of quotients of the Fermat curve \(x^p+y^p=1\), for certain primes \(p\). His formula involved finding a \(p\)-adic approximation to a certain function, instead of local Hilbert norm residue symbols. \textit{J. W. S. Cassels} [J. Reine Angew. Math. 494, 101-127 (1998; Zbl 0883.11028)] gives an explicit, classical approach to higher 2-power descents. \textit{T.~Fisher} [J. Eur. Math. Soc. 3, 169-201 (2001; Zbl 1007.11031)] proves that the \(5\)-torsion of Shafarevich-Tate groups of elliptic curves over \(\mathbb Q\) can be arbitrarily large and performs some calculations for \(7\)-descents as well. Also the recent preprint ``The Cassels-Tate pairing and the Platonic solids'' of Fisher extends the results of the paper under review.
scientific article; zbMATH DE number 1538252
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