The Šafarevič-Tate group in cyclotomic \(\mathbb Z_p\)-extensions at supersingular primes (Q2848381)

The main goal of this paper is the study of the asymptotic growth of the \(p\)--primary component of the Shafarevich-Tate group in the cyclotomic direction at any odd prime of good supersingular reduction. Not much is known about its size and how it behaves according to the number field. \textit{B. Mazur} [Invent. Math. 18, 183--266 (1972; Zbl 0245.14015)] studied the case of a \({\mathbb Z}_p\)-extension when \(p\) has good ordinary reduction. That is, let \(E\) be an elliptic curve over \({\mathbb Q}\) and \(p\) an odd prime of good reduction and let \({\mathbb Q}_{\infty}\) be the cyclotomic \({\mathbb Z}_p\)-extension, \(\Gamma:=\mathrm{Gal}({\mathbb Q}_{\infty}/{\mathbb Q}) \cong {\mathbb Z}_p\). Let \(\text Ш(E/{\mathbb Q}_n):=\ker \big(H^1({\mathbb Q}_n,E)\to \prod_v H^1({\mathbb Q}_{n,v},E)\big)\) be the Shafarevich-Tate group at level \(n\), where \(v\) varies over all places of \({\mathbb Q}_n\) and let \(p^{e_n}\) be its order. When \(p\) is ordinary, Mazur found [loc. cit.] that NEWLINE\[NEWLINE e_n-e_{n-1}=(p^n-p^{n-1})\mu +\lambda-r_{\infty} NEWLINE\]NEWLINE for \(n\gg 0\) and where \(r_{\infty}\) is the rank of \(E({\mathbb Q}_{\infty})\) which is an analogue of the result of Iwasawa for the class number of the \(n\)-th layer \({\mathbb Q}_n\) of \({\mathbb Q}_{\infty}/{\mathbb Q}\).NEWLINENEWLINEThe supersingular case, that is, when \(p\) divides \(a_p=p+1-\#E({ \mathbb F}_p)\) was studied by \textit{M. Kurihara} [Invent. Math. 149, No. 1, 195--224 (2002; Zbl 1033.11028)] and by \textit{B. Perrin-Riou} [Exp. Math. 12, No. 2, 155--186 (2003; Zbl 1061.11031)]. The case \(a_p=0\) has been more accessible. The general supersingular case has turned out to be more difficult. In this paper, the author provides three possible growth formulas involving algebraic Iwasawa invariants coming from two modified Selmer groups. He also generalizes Perrin-Riou's work giving the growth pattern of the Shafarevich-Tate group in the tower of number fields obtained by adjoining all \(p\)-powers roots of unity to \({\mathbb Q}\).NEWLINENEWLINENEWLINEThe main result in this work states that if \(p\) is an odd supersingular prime and \(\text Ш(E/{\mathbb Q}_n)[p^{\infty}]\) is finite, there exist integer invariants \(\mu_{\sharp}\), \(\lambda_{\sharp}\), \(\mu_{\flat}\), \(\lambda_{\flat}\) so that for \(n\gg 0\), NEWLINE\[NEWLINE e_n-e_{n-1}=\left\{\begin{matrix} (p^n-p^{n-1})\mu_{\sharp}+\lambda_{\sharp}-r_{\infty}+q_n^{\sharp} \text{\;if \(n\) is even}\cr (p^n-p^{n-1})\mu_{\flat}+\lambda_{\flat}-r_{\infty}+q_n^{\flat} \text{\;if \(n\) is odd}\end{matrix}\right. NEWLINE\]NEWLINE when \(a_p=0\) or \(\mu_{\sharp}=\mu_{\flat}\) and NEWLINE\[NEWLINE e_n-e_{n-1}=(p^n-p^{n-1})\mu_{\ast}+\lambda_{\ast}-r_{\infty}+q_n^{\ast} NEWLINE\]NEWLINE when \(a_p\neq 0\) and \(\mu_{\sharp}\neq \mu_{\flat}\), \(\ast\in\{\sharp,\flat\}\) is chosen so that \(\mu_{\ast}=\min\{\mu_{\sharp},\mu_{\flat}\}\). Here \(q_n^{\sharp}\) and \(q_n^{\flat}\) are explicitly given functions on \(p\) and \(n\).