##
**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)].

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)].

Reviewer: Andreas Nickel (Essen)

### Keywords:

Iwasawa main conjecture; Hida theory; Selmer groups; Heegner points; modular forms; Iwasawa invariants; special \(L\)-values### Citations:

Zbl 1259.11101; Zbl 0882.11034; Zbl 1301.11074; Zbl 1326.11067; Zbl 1093.11065; Zbl 1093.11037; Zbl 1320.11054; Zbl 1436.11053; Zbl 1408.11022
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\textit{F. Castella} et al., Algebra Number Theory 11, No. 10, 2339--2368 (2017; Zbl 1404.11126)

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