The enigmatic Tate-Shafarevich group (Q2900275)

In this expository article, J. H. Coates discusses what is currently known and conjectured about the Tate-Shafarevich group of an elliptic curve defined over the rational numbers.NEWLINENEWLINEGiven an elliptic curve \(E\) defined over \(\mathbb{Q}\), the Tate-Shafarevich group of \(E\) is defined by NEWLINE\[NEWLINE \text{Ш} (E/\mathbb{Q}) = \mathrm{Ker} \Big( H^1(\mathbb{Q},E) \rightarrow \bigoplus_v H^1(\mathbb{Q}_v,E) \Big) . NEWLINE\]NEWLINE As the author writes in section 1, there is a large gap between what is known about this group and what is conjectured. The Tate-Shafarevich group is conjecturally always finite, but this has not been established for any example of \(E\) having rank at least 2. However, the Birch and Swinnerton-Dyer Conjecture gives a formula for the order of the group, which predicts that it is small (or even trivial) in many cases. The author illustrates this with an example of a curve having rank 3 for which this formula predicts \(\text{Ш} (E/\mathbb{Q}) = 0\).NEWLINENEWLINEIn section 2 the author explains how the Tate-Shafarevich group is connected to the different \(L\)-functions attached to \(E\). The first example is the Birch and Swinnerton-Dyer Conjecture, which involves the complex \(L\)-function \(L(E,s)\). Again, little is known about this when the rank of \(E\) is at least 2 and the author turns to \(p\)-adic analogues of this conjecture to make further progress.NEWLINENEWLINEWhen the elliptic curve \(E\) has no complex multiplication, one can obtain some results (for which references are given in the paper) using the Mazur-Swinnerton-Dyer \(p\)-adic \(L\)-functions attached to the cyclotomic \(\mathbb{Z}_p\)-extension of \(\mathbb{Q}\). However, when \(E\) admits complex multiplication and has good ordinary reduction at \(p\), one can do better using a \(p\)-adic \(L\)-function \(L_{\mathfrak{p}}(E,s)\) attached to a \(\mathbb{Z}_p\)-extension of the imaginary quadratic field. In \textit{J. Coates, Z. Liang} and \textit{R. Sujatha} [Milan J. Math. 78, No. 2, 395--416 (2010; Zbl 1295.11059)] an upper bound for the order of vanishing for \(L_{\mathfrak{p}}(E,s)\) at \(s=1\) is established when \(p\) is sufficiently large. Combining this with the main conjecture of Iwasawa theory for imaginary quadratic fields (see \textit{K. Rubin} [Invent. Math. 103, No. 1, 25--68 (1991; Zbl 0737.11030)]) the author deduces a theoretical result on the \(p\)-primary part of the Tate-Shafarevich group.NEWLINENEWLINENumerical results are also discussed in the article: for several example elliptic curves with complex multiplication by \(\mathbb{Z}[i]\), the finiteness of the \(p\)-primary part \(\text{Ш} (E/\mathbb{Q})(p)\) has been computationally verified for all \(p \equiv 1 \,\mathrm{mod} \, 4\) in the range \(p<30\,000\) (see \textit{J. Coates, Z. Liang} and \textit{R. Sujatha} [J. Algebra 322, No. 3, 657--674 (2009; Zbl 1239.11069)] and \textit{J. Coates, Z. Liang} and \textit{R. Sujatha} (loc. cit.). The calculations (with work of C. Wuthrich and Z. Liang to deal with exceptional cases) show that \(\text{Ш} (E/\mathbb{Q})(p)=0\) in these cases, which is consistent with the Birch and Swinnerton-Dyer Conjecture. In fact, they establish something stronger about the characteristic power series of the dual of the \(\mathfrak{p}^\infty\)-Selmer group for these examples (see the end of section 2).NEWLINENEWLINEIn section 3, the author states some speculative conjectures about abelian varieties over the compositum of different \(\mathbb{Z}_p\)-extensions. To be more precise, given a number field \(F\) we write \(F(cyc)\) for the compositum of the cyclotomic \(\mathbb{Z}_p\)-extensions of \(F\) for all primes \(p\). The author discusses conjectures about the arithmetic of an abelian variety over \(F(cyc)\), including a generalisation of Rohrlich's non-vanishing theorem (see \textit{D. E. Rohrlich} [Invent. Math. 75, 409--423 (1984; Zbl 0565.14006)]). He also discusses a conjecture on class numbers of subfields of \(F(cyc)\) with some numerical results in this direction.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].