×

Estimating the growth in Mordell-Weil ranks and Shafarevich-Tate groups over Lie extensions. (English) Zbl 1431.11122

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) and let \(p>3\) be a prime, where \(E\) has good ordinary reduction. Let \(k\) be a number field. Under certain hypotheses, the authors study the growth of the Mordell-Weil ranks and Shafarevich-Tate groups of \(E\) over towers of extensions \(K_n/k\) in a \(p\)-adic Lie extension \(K_{\infty}\) of \(k\).
The first part of the paper is devoted to a generalization of a result on the Mordell-Weil ranks due to H. Darmon and Y. Tian [Can. J. Math. 62, No. 5, 1060–1081 (2010; Zbl 1287.11081)]. More precisely, they give upper and lower bounds for \(p\)-adic Lie extensions of number fields whose Galois group is \(\mathbb Z_p^2 \rtimes \mathbb Z_p\), and for \((d-1)\)-fold false Tate extensions \(K_{\infty} = \mathbb Q(\mu_{p^{\infty}}, \ell_1^{1 / p^{\infty}}, \dots, \ell_{d-1}^{1 / p^{\infty}})\). In certain special cases, even the exact rank is determined.
In the second part, the authors examine the growth of the \(p\)-primary parts of Shafarevich-Tate groups (under the assumption that they are finite). They obtain an asymptotic upper bound, using the lower bound for Mordell-Weil ranks of the first part and a refinement of R. Greenberg’s control theorem [Compos. Math. 136, No. 3, 255–297 (2003; Zbl 1158.11319)].
These formulae are reminiscent to corresponding formulae for class groups in \(\mathbb Z_p^d\)-extensions due to A. A. Cuoco and P. Monsky [Math. Ann. 255, 235–258 (1981; Zbl 0437.12003)] and also in non-commutative extensions due to G. Perbet [Algebra Number Theory 5, No. 6, 819–848 (2011; Zbl 1275.11142)].

MSC:

11R23 Iwasawa theory
11G05 Elliptic curves over global fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Darmon, H.: Tian, Ye: Heegner points over towers of Kummer extensions. Can. J. Math. 62(5), 1060-1081 (2010) · Zbl 1287.11081 · doi:10.4153/CJM-2010-039-8
[2] Dokchitser, T.: Dokchitser, Vladimir: Regulator constants and the parity conjecture. Invent. Math. 178(1), 23-71 (2009) · Zbl 1219.11083 · doi:10.1007/s00222-009-0193-7
[3] Greenberg, R.: Iwasawa Theory, Projective Modules, and Modular Representations, vol. 211, no. 992. Memoirs of the American Mathematical Society (2011) · Zbl 1247.11085
[4] Guo, L.: General Selmer groups and critical values of Hecke \[L\] L-functions. Math. Ann. 297(2), 221-233 (1993) · Zbl 0789.14018 · doi:10.1007/BF01459498
[5] Greenberg, R.: Galois theory for the Selmer group of an abelian variety. Compos. Math. 136(3), 255-297 (2003) · Zbl 1158.11319 · doi:10.1023/A:1023251032273
[6] Cuoco, A.A., Monsky, P.: Class numbers in \[{\mathbf{Z}}^d_p\] Zpd-extensions. Math. Ann. 255(2), 235-258 (1981) · Zbl 0437.12003 · doi:10.1007/BF01450674
[7] Perbet, G.: Sur les invariants d’Iwasawa dans les extensions de Lie \[p\] p-adiques. Algebr. Number Theory 5(6), 819-848 (2011) · Zbl 1275.11142 · doi:10.2140/ant.2011.5.819
[8] Gonzalez-Sanchez, J., Klopsch, B.: Analytic pro-p groups of small dimensions. J. Group Theory 12(5), 711-734 (2009) · Zbl 1183.20030 · doi:10.1515/JGT.2009.006
[9] Delbourgo, D., Lei, A.: Transition formulae for ranks of abelian varieties. Rocky Mt. J. Math. 45(6), 1807-1838 (2015) · Zbl 1355.11070 · doi:10.1216/RMJ-2015-45-6-1807
[10] Coates, J., Fukaya, T., Kato, K., Sujatha, R.: Root numbers, Selmer groups, and non-commutative Iwasawa theory. J. Algebr. Geom. 19(1), 19-97 (2010) · Zbl 1213.11135 · doi:10.1090/S1056-3911-09-00504-9
[11] Dokchitser, V.: Root numbers of non-abelian twists of elliptic curves. Proc. Lond. Math. Soc. (3) 91, 300-324 (2005). With an appendix by Tom Fisher · Zbl 1076.11042 · doi:10.1112/S0024611505015261
[12] Delbourgo, D.: Peters, Lloyd: Higher order congruences amongst Hasse-Weil \[L\] L-values. J. Aust. Math. Soc. 98(1), 1-38 (2015) · Zbl 1348.11084 · doi:10.1017/S1446788714000445
[13] Zhang, S.W.: Gross-Zagier Formula for \[{{\rm GL}}(2)\] GL(2). II, Heegner Points and Rankin \[L\] L-Series. Mathematical Sciences Research Institute Publications. Cambridge University Press, Cambridge (2004)
[14] Skinner, C., Urban, E.: The Iwasawa main conjectures for \[{{\rm GL}}_2\] GL2. Invent. Math. 195(1), 1-277 (2014) · Zbl 1301.11074 · doi:10.1007/s00222-013-0448-1
[15] Zerbes, S.L.: Selmer groups over \[p\] p-adic Lie extensions. J. Lond. Math. Soc. 70(3), 586-608 (2004) · Zbl 1065.11039 · doi:10.1112/S002461070400568X
[16] Harris, M.: \[p\] p-Adic representations arising from descent on abelian varieties. Compos. Math. 39(2), 177-245 (1979) · Zbl 0417.14034
[17] Coates, J.: Fragments of the \[{{\rm GL}}_2\] GL2 Iwasawa theory of elliptic curves without complex multiplication. In: Arithmetic Theory of Elliptic Curves (Cetraro, 1997). Lecture Notes in Mathematics, vol. 1716, pp. 1-50. Springer, Berlin (1999) · Zbl 1029.11016
[18] Serre, J.P.: Cohomologie Galoisienne, Cours au Collège de France, vol. 1962. Springer, Berlin (1962/1963) · Zbl 0145.17501
[19] Hachimori, Y., Venjakob, O.: Completely faithful Selmer groups over Kummer extensions. Kazuya Kato’s fiftieth birthday. Doc. Math., Extra vol., 443-478 (2003) (electronic) · Zbl 1117.14046
[20] Burns, D., Venjakob, O.: On descent theory and main conjectures in non-commutative Iwasawa theory. J. Inst. Math. Jussieu 10(1), 59-118 (2011) · Zbl 1213.11134 · doi:10.1017/S147474800900022X
[21] Howson, S.: Euler characteristics as invariants of Iwasawa modules. Proc. Lond. Math. Soc. 3(85), 634-658 (2002) · Zbl 1036.11053 · doi:10.1112/S0024611502013680
[22] Venjakob, O.: On the structure theory of the Iwasawa algebra of a \[p\] p-adic Lie group. J. Eur. Math. Soc. (JEMS) 4(3), 271-311 (2002) · Zbl 1049.16016 · doi:10.1007/s100970100038
[23] Ritter, J., Weiss, A.: Toward equivariant Iwasawa theory. III. Math. Ann. 336(1), 27-49 (2006) · Zbl 1154.11038 · doi:10.1007/s00208-006-0773-4
[24] Bourbaki, N.: Éléments de mathématique. Fasc. XXXI. Algèbre commutative. Chapitre 7: Diviseurs, Actualités Scientifiques et Industrielles, vol. 1314. Hermann, Paris (1965) · Zbl 0141.03501
[25] Kobayashi, S.: Iwasawa theory for elliptic curves at supersingular primes. Invent. Math. 152(1), 1-36 (2003) · Zbl 1047.11105 · doi:10.1007/s00222-002-0265-4
[26] Coates, J.: Elliptic Curves—The Crossroads of Theory and Computation. Lecture Notes in Computer Science, vol. 2369, pp. 9-19 (2002) · Zbl 1133.11311
[27] Matsuno, K.: Finite \[\Lambda\] Λ-submodules of Selmer groups of abelian varieties over cyclotomic \[{\mathbb{Z}}_p\] Zp-extensions. J. Number Theory 99(2), 415-443 (2003) · Zbl 1045.11042 · doi:10.1016/S0022-314X(02)00078-1
[28] Coates, J., Schneider, P., Sujatha, R.: Links between cyclotomic and \[{\text{ GL }}_2\] GL2 Iwasawa theory. Kazuya Kato’s fiftieth birthday. Doc. Math. Extra vol., 187-215 (2003) (electronic) · Zbl 1142.11366
[29] Delbourgo, D., Lei, A.: Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction. Math. Proc. Cambridge Philos. Soc. 160(1), 11-38 (2016) · Zbl 1371.11142 · doi:10.1017/S0305004115000535
[30] Venjakob, O.: A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math. 559, 153-191 (2003). With an appendix by Denis Vogel · Zbl 1051.11056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.