Elliptic curves with large Tate-Shafarevich groups over a number field
In this nice paper the author studies the unboundedness of the Tate-Shafarevich group in a family of elliptic curves defined over a fixed number field. The problem of determining how big this group can be for various elliptic curves is classical and has extensively been studied by Cassels, Kloosterman, Kramer and many others (see for instance \textit{J. W. S. Cassels} [J. Reine Angew. Math. 214--215, 65--70 (1964; Zbl 0236.14012)], \textit{R. Kloosterman} [J. Théor. Nombres Bordx. 17, No. 3, 787--800 (2005; Zbl 1153.11313)], \textit{K. Kramer} [Proc. Am. Math. Soc. 89, 379--386 (1983; Zbl 0567.14018)] and references there). A refinement of the above question is to study, for a fixed prime \(p\), the unboundedness of the \(p\)-part of the Tate-Shafarevich group. In this direction, for \(p=3\), Cassels gave a positive answer in the paper cited above showing that the \(3\)-part of \(\text Ш(E/\mathbb{Q})\) is unbounded if \(E\) runs through all elliptic curves with \(j\)-invariant \(0\). This result was eventually generalized in [\textit{N. Aoki}, Acta Arith. 112, No. 3, 209--227 (2004; Zbl 1083.11035); \textit{R. Bölling}, Math. Nachr. 67, 157--179 (1975; Zbl 0314.14008); \textit{T. Fisher}, J. Eur. Math. Soc. (JEMS) 3, No. 2, 169--201 (2001; Zbl 1007.11031); \textit{R. Kloosterman} and \textit{E. F. Schaefer}, J. Number Theory 99, No. 1, 148--163 (2003; Zbl 1074.11032); \textit{K. Matsuno}, Manuscr. Math. 122, No. 3, 289--304 (2007; Zbl 1152.11045)] to \(p\leq 7\) and \(p=13\). The main technique employed there was to produce infinitely many elliptic curves over \(\mathbb{Q}\) with a \(p\)-isogeny and to study the effect of these isogenies on the Tate-Shafarevich group. Since the genus of \(X_0(p)\) is greater than \(1\) for \(p\geq 23\), for such primes there are only finitely many elliptic curves over \(\mathbb{Q}\) with a \(p\)-isogeny defined over \(\mathbb{Q}\); therefore the above strategy breaks down for large primes. Observe that this obstruction occurs also for any general number field \(K/\mathbb{Q}\), but if one allows \(K\) to vary then it can be shown [see Kloosterman, 2005 (loc.cit.)] that the \(p\)-part of the Tate-Shafarevich group of \(E/K\), for \(K\) varying over number fields and \(E/K\) over all elliptic curves over \(K\), can be arbitrarily large. This result can therefore be strengthened in two ways: either fixing \(E\) (defined, say, over \(\mathbb{Q}\)) and letting \(K\) vary over all possible number fields, or fixing \(K\) and letting \(E/\mathbb{Q}\) vary over all possible elliptic curves. The first approach has been investigated in a preprint by \textit{P. L. Clark} and \textit{S. Sharif} [published in Algebra Number Theory 4, No. 2, 151--174 (2010; Zbl 1200.11037)] and shows that for every elliptic curve \(E/\mathbb{Q}\), the \(p\)-part of \(\text Ш(E/K)\) is unbounded when \(K\) runs through all number fields of degree \(p\). The paper under discussion considers the other case, thus fixing once and for all a cyclic extension \(K/\mathbb{Q}\) of degree \(p\) and considering \(\text Ш(E/K)\) for \(E/\mathbb{Q}\) running through all elliptic curves defined over \(\mathbb{Q}\); the author proves that the \(p\)-part of this group is unbounded, and actually exhibits an ``explicit'' family of curves for which this happens (there is a non-explicit step in the construction of this family, coming from sieve theory, but this can be overcome in concrete examples). The paper falls in six sections: after an Introduction and a notational section, in Section 3 the author presents some classical cohomological consequence of global duality to show that the rank of the Selmer group can be bounded from below by the rank of a certain cohomology group \(W_{v,K}\subseteq H^1(\mathbb{Q}_v,E(\overline{\mathbb{Q}_v}))\), for all primes \(v\) of \(K\) in a finite set of \textit{bad} primes for \(E\) and for \(K/\mathbb{Q}\). After this preliminary result, the strategy is to find in Section 4 criteria under which this \(W_{v,K}\) is ``big'' and in Section 5 criteria under which \(E(K)\) has ``small'' rank: the comparison of the two estimates leads, via the exact sequence defining the Tate-Shafarevich group, to a lower bound for its size. Finally, in Section 6, the case \(p=2\) is discussed, because it requires a quadratic-twist procedure that is not needed for odd primes; in this last section, the author provides also for a new proof of the mentioned result by Clark and Sharif in case \(p=2\). In more details, the main result of Section 4. is Proposition 4.3 below: let \(K/\mathbb{Q}\) be a cyclic extension of degree \(n\) (eventually the author considers \(n=p\)) and let \(E/\mathbb{Q}\) be an elliptic curve. Then two technical properties -- say \textbf{P1} and \textbf{P2}, depending on \(E\) and \(K\) -- for a prime number \(\ell\nmid n\) are defined; let \(T_{E,K}\) be a finite set of primes verifying either \textbf{P1} or \textbf{P2}, and set \(t_{E,K}=|T_{E,K}|\). Then Proposition 4.3: For \(E\) and \(K\) as above, we have \[ \mathrm{rk}_n\mathrm{Sel}_n(E/K)\geq t_{E,K}-2\max_{p\mid n}\{\mathrm{rk}_pE(\mathbb{Q})[p]\}-\delta'\geq t_{E,K}-4 \] where, for an integer \(n\) and an abelian group \(A\), we denote by \(\mathrm{rk}_nA\) the largest \(r\) such that \(A\) admits a subgroup isomorphic to \(r\) copies of the cyclic group of order \(n\); and where \(\delta'=1\) if \(n\) is even and \(\text{rk}_2E(\mathbb{Q})[2]=1\), and \(\delta'=0\) if not. The proof of the Proposition is a simple cohomological computation, where properties \textbf{P1} and \textbf{P2} become conditions on the Galois representation attached to the \(p\)-torsion of \(E\). Section 5 deals with a upper bound for the \(K\)-rank of suitable elliptic curves defined over \(\mathbb{Q}\); it is deeply based on the techniques introduced by Kramer in the paper cited above who studies elliptic curves defined by \[ A(m,l):y^2+xy=x^3+8mx^2+lmx\tag{A} \] for suitable \(m\) and \(l\). Thanks to an intriguing argument involving classical algebraic number theory and sieve methods, the author constructs for every parameter \(k\in\mathbb{N}\) two integers \(m_k\) and \(l_k\) verifying Kramer's hypothesis and is able to prove the following Corollary 5.5: Assume \(K\) is as above and \(A/\mathbb{Q}\) is given by equation (A). Then \[ \mathrm{rank}_\mathbb{Z}A(m_k,l_k)(K)\leq 14n+2h-2 \] where \(h\) is the \(2\)-rank of the \(\Sigma\)-ideal class group of \(K\) for \(\Sigma=\{v\mid \infty\}\cup\{v\mid 2m_kl_k\}\) (in particular, thus, \(h\leq \mathrm{rk}_2\mathrm{Cl}_K\)). For odd \(n\), the explicit knowledge of the curve \(A(m_k,l_k)\) found in Kramer's work cited above allows the author to show in the proof of Corollary 5.6 that there exists a set \(T_{A,K}\) of primes verifying either \textbf{P1} or \textbf{P2} for \(A(m_k,l_k)\) with \(t_{A,K}\geq k-5\): combining Proposition 4.3 and Corollary 5.5 above he can therefore produce, for every \(k\), an elliptic curve over \(K\) with a Tate-Shafarevich group \(\text Ш(E/K)\) whose \(n\)-rank is at least \(k\). The paper ends with Section 6, where the special case \(p=2\) is treated.
scientific article; zbMATH DE number 5608655