Variation of Iwasawa invariants in Hida families. (English) Zbl 1093.11065

Consider absolutely irreducible modular Galois representations \(\overline\rho: G_\mathbb{Q}\to GL_2(k)\), where \(k\) is a finite field of characteristic \(p\), and assume that \(\overline\rho\) is \(p\)-ordinary and that its restriction to a decomposition group at \(p\) is reducible and non-scalar. The corresponding Hida family \({\mathcal H}(\overline\rho)\) is the set of all \(p\)-ordinary \(p\)-stabilized newforms \(f\) with mod],\(p\) Galois representation isomorphic to \(\overline\rho\). Each \(f\) gives rise to the analytic and algebraic Iwasawa invariants \(\mu^{\text{an}}, \lambda^{\text{an}},\mu^{\text{alg}},\lambda^{\text{alg}}\). Theorem 1. If \(\mu^*(f_0)=0\) for some \(f_0\in{\mathcal H}(\overline\rho)\), then \(\mu^*(f)=0\) for all \(f\in{\mathcal H}(\overline\rho)\) (i.e., \(\mu^* (\overline\rho)=0)\); here \(*\) is ‘an’ or ‘alg’. It follows that Greenberg’s conjecture, \(\mu^{\text{an}}(f)= \mu^{\text{alg}}(f)=0\) for \(p\)-ordinary modular \(f\) of weight two having a residually irreducible Galois representation, is equivalent to the corresponding conjecture without the weight restriction. Theorem 2. Assume that \(\mu^* (\overline\rho)=0\) and let \(f_{1,2}\in{\mathcal H}(\overline\rho)\) lie on the branches \({\mathbf T}({\mathfrak a}_{1,2})\). Then \(\lambda^*(f_1)-\lambda^*(f_2)=\sum e_\ell({\mathfrak a}_2)-e_ell({\gamma a}_1)\) with the sum taken over all primes \(\ell\) dividing the tame level of \(f_1\) or \(f_2\), and with \(e_\ell ({\mathfrak a})\) an explicit non-negative invariant of the branch \({\mathbf T} ({\mathfrak a})\) and the prime \(\ell\) \(({\mathfrak a}\) is a minimal prime ideal of a certain Hecke algebra.) Corollaries. 1. \(\mu^{\text{an}}(\overline \rho)=\mu^{\text{alg}} (\overline\rho) =0\) implies the validity of the main conjecture for all \(f\in{\mathcal H}(\overline \rho)\), if it only holds for some \(f_0\in{\mathcal H}(\overline\rho)\). 2. If \(\mu^* (\overline\rho)=0\), then \(\lambda^*\) is constant on branches of \({\mathcal H} (\overline\rho)\) and minimized on the branches of minimal tame level. The proofs of the theorems are based on, for the analytic part, two-variable \(p\)-adic \(L\)-functions on the Hida family of \(\overline\rho\) and on each of its irreducible components and, for the algebraic part, the theory of residual Selmer groups and the use of level lowering. Applications to the main conjecture and examples finish the paper.


11R23 Iwasawa theory
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