Variation of anticyclotomic Iwasawa invariants in Hida families. (English) Zbl 1404.11126

Building upon the known cases of the anticyclotomic Iwasawa main conjecture in weight \(2\), the authors prove new cases of the main conjecture for \(p\)-ordinary newforms of higher weights and trivial nebentypus.
Let \(p>3\) be a prime and let \(K\) be an imaginary quadratic field such that \(p\) splits in \(K\). Let \(K_{\infty}\) be the anticyclotomic \(\mathbb Z_p\)-extension of \(K\) and write \(\Lambda\) for the associated Iwasawa algebra. Consider the Hida family \(\mathcal{H}(\overline \rho)\) of all \(p\)-ordinary and \(p\)-stabilized newforms with mod \(p\) Galois representation isomorphic to some fixed irreducible Galois representation \(\overline \rho: G_{\mathbb Q} \rightarrow \mathrm{GL}_2(\mathbb F)\). Following the terminology of R. Pollack and T. Weston [Compos. Math. 147, No. 5, 1353–1381 (2011; Zbl 1259.11101)] the authors define a minimal Selmer group \(\mathrm{Sel}(K_{\infty},f)\) for each \(f \in \mathcal{H}(\overline \rho)\). This Selmer group is known to be cofinitely generated and expected to be \(\Lambda\)-cotorsion. The main conjecture in this setting then asserts that its characteristic ideal is generated by an associated \(p\)-adic \(L\)-function \(L_p(f/K) \in \Lambda\) constructed by M. Bertolini and H. Darmon [Invent. Math. 126, No. 3, 413–456 (1996; Zbl 0882.11034)] in weight two and more recently by M. Chida and M.-L. Hsieh [J. Reine Angew. Math. 741, 87–131 (2018; Zbl 1436.11053)] for heigher weights.
One divisibility (under certain hypotheses that we are always going to suppress in this review) follows from the work of C. Skinner and E. Urban [Invent. Math. 195, No. 1, 1–277 (2014; Zbl 1301.11074)] on the three-variable main conjecture. The other divisibility is known for newforms \(f\) of weight \(k \leq p-2\) and trivial nebentypus again by work of M. Chida and M.-L. Hsieh [Compos. Math. 151, No. 5, 863–897 (2015; Zbl 1326.11067)].
However, their arguments do not generalize to higher weights and the authors of the present article follow an approach by M. Emerton, R. Pollack and T. Weston [Invent. Math. 163, No. 3, 523–580 (2006; Zbl 1093.11065)] in the cyclotomic setting to verify the anticyclotomic main conjecture for all \(f\) of weight \(k \equiv 2 \,\mathrm{mod}\, p-1\) and trivial nebentypus. More precisely, they show that the \(\mu\) and \(\lambda\)-invariants of the dual minimal Selmer group and \(L_p(f,K)\) vary in the same way. This reduces the problem to the weight two case, where the results of M. Bertolini and H. Darmon [Ann. Math. (2) 162, No. 1, 1–64 (2005; Zbl 1093.11037)] and Pollack and Weston [loc. cit.] apply.
On the algebraic side the arguments of Emerton et al. [loc. cit.] carry over almost unchanged. On the analytic side, however, they cannot work with classical modular curves, but with a family of Shimura curves associated to definite quarternion algebras. The authors build upon their work with S. Vigni [Manuscr. Math. 135, No. 3–4, 273–328 (2011; Zbl 1320.11054)] and [Ann. Math. Qué. 40, No. 2, 303–324 (2016; Zbl 1408.11022)].


11R23 Iwasawa theory
11F33 Congruences for modular and \(p\)-adic modular forms
Full Text: DOI arXiv


[1] 10.1007/s002220050105 · Zbl 0882.11034
[2] 10.4007/annals.2005.162.1 · Zbl 1093.11037
[3] 10.1007/s40316-015-0045-3 · Zbl 1408.11022
[4] 10.1112/S0010437X14007787 · Zbl 1326.11067
[5] 10.1515/crelle-2015-0072 · Zbl 1436.11053
[6] 10.1007/BF01231768 · Zbl 0847.11025
[7] 10.1007/s00222-005-0467-7 · Zbl 1093.11065
[8] ; Greenberg, Algebraic number theory. Adv. Stud. Pure Math., 17, 97 (1989) · Zbl 0739.11045
[9] 10.1007/s002220000080 · Zbl 1032.11046
[10] 10.1007/BF01390329 · Zbl 0612.10021
[11] 10.2307/1971444 · Zbl 0658.10034
[12] 10.1007/s00222-006-0007-0 · Zbl 1171.11033
[13] 10.1007/s00222-009-0205-7 · Zbl 1304.11041
[14] 10.4310/AJM.2017.v21.n3.a5 · Zbl 1441.11276
[15] 10.1142/S1793042117500804 · Zbl 1428.11193
[16] 10.1007/s00229-010-0409-6 · Zbl 1320.11054
[17] 10.1112/S0010437X11005318 · Zbl 1259.11101
[18] 10.1515/9781400865208 · Zbl 0977.11001
[19] 10.1007/s00222-013-0448-1 · Zbl 1301.11074
[20] ; Skinner, Inst. Hautes Études Sci. Publ. Math., 5 (1999)
[21] 10.1215/S0012-7094-03-11622-1 · Zbl 1065.11048
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.