On the anticyclotomic Iwasawa main conjecture for modular forms. (English) Zbl 1326.11067
In the paper under review the authors generalize the work of M. Bertolini and H. Darmon [Ann. Math. (2) 162, No. 1, 1–64 (2005; Zbl 1093.11037)] on the anticyclotomic main conjecture for elliptic curves to modular forms of higher weight.
More precisely, let \(f \in S_k(\Gamma_0(N))\) be an elliptic newform of level \(N\). Let \(K\) be an imaginary quadratic field with discriminant \(D_K\). As usual one decomposes \(N\) as \(N = N^+ N^-\), where \(N^+\) is only divisible by primes split in \(K\) and \(N^-\) is only divisible by primes which are inert or ramified in \(K\). The authors assume that \(N^-\) is a square-free product of an odd number of inert primes.
Let \(p \nmid N D_K\) be a prime and suppose that \(f\) is \(p\)-ordinary. One can associate to \(f\) a certain two-dimensional (over the Hecke field of \(f\)) self-dual Galois representation \(V_f\) and a (minimal) Selmer group \(\mathrm{Sel}(K_{\infty}, V_f / T_f)\), where \(K_{\infty}\) is the anticyclotomic \(\mathbb Z_p\)-extension of \(K\) and \(T_f \subset V_f\) is a Galois stable lattice. The Pontryagin dual \(\mathrm{Sel}(K_{\infty}, V_f / T_f)^{\vee}\) is known to be a finitely generated Iwaswa module.
The main result of this article now states that the characteristic ideal of \(\mathrm{Sel}(K_{\infty}, V_f / T_f)^{\vee}\) contains the \(p\)-adic \(L\)-series \(L_p(K_{\infty}, f)\) under certain hypotheses. As a corollary, one obtains that the \(\mu\)-invariant of \(\mathrm{Sel}(K_{\infty}, V_f / T_f)^{\vee}\) vanishes in this situation. The \(p\)-adic \(L\)-series has been considered in the author’s article [“Special values of anticyclotomic \(L\)-functions for modular forms”, Preprint, arXiv:1204.2427], where almost the same hypotheses already appear. Note that the case \(k=2\) of the main theorem recovers the result of Bertolini and Darmon mentioned above.
The prove heavily relies on the existence of a certain Euler system constructed via level-raising. The heart of the paper is then devoted to establishing a connection between this Euler system an the \(p\)-adic \(L\)-function.
As the authors point out it seems to be likely that the other divisibility follows from work of C. Skinner and E. Urban [Invent. Math. 195, No. 1, 1–277 (2014; Zbl 1301.11074)].
More precisely, let \(f \in S_k(\Gamma_0(N))\) be an elliptic newform of level \(N\). Let \(K\) be an imaginary quadratic field with discriminant \(D_K\). As usual one decomposes \(N\) as \(N = N^+ N^-\), where \(N^+\) is only divisible by primes split in \(K\) and \(N^-\) is only divisible by primes which are inert or ramified in \(K\). The authors assume that \(N^-\) is a square-free product of an odd number of inert primes.
Let \(p \nmid N D_K\) be a prime and suppose that \(f\) is \(p\)-ordinary. One can associate to \(f\) a certain two-dimensional (over the Hecke field of \(f\)) self-dual Galois representation \(V_f\) and a (minimal) Selmer group \(\mathrm{Sel}(K_{\infty}, V_f / T_f)\), where \(K_{\infty}\) is the anticyclotomic \(\mathbb Z_p\)-extension of \(K\) and \(T_f \subset V_f\) is a Galois stable lattice. The Pontryagin dual \(\mathrm{Sel}(K_{\infty}, V_f / T_f)^{\vee}\) is known to be a finitely generated Iwaswa module.
The main result of this article now states that the characteristic ideal of \(\mathrm{Sel}(K_{\infty}, V_f / T_f)^{\vee}\) contains the \(p\)-adic \(L\)-series \(L_p(K_{\infty}, f)\) under certain hypotheses. As a corollary, one obtains that the \(\mu\)-invariant of \(\mathrm{Sel}(K_{\infty}, V_f / T_f)^{\vee}\) vanishes in this situation. The \(p\)-adic \(L\)-series has been considered in the author’s article [“Special values of anticyclotomic \(L\)-functions for modular forms”, Preprint, arXiv:1204.2427], where almost the same hypotheses already appear. Note that the case \(k=2\) of the main theorem recovers the result of Bertolini and Darmon mentioned above.
The prove heavily relies on the existence of a certain Euler system constructed via level-raising. The heart of the paper is then devoted to establishing a connection between this Euler system an the \(p\)-adic \(L\)-function.
As the authors point out it seems to be likely that the other divisibility follows from work of C. Skinner and E. Urban [Invent. Math. 195, No. 1, 1–277 (2014; Zbl 1301.11074)].
Reviewer: Andreas Nickel (Bielefeld)
MSC:
| 11R23 | Iwasawa theory |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
References:
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