Congruences among modular forms on \(\text{U}(2,2)\) and the Bloch-Kato conjecture. (English) Zbl 1214.11055
Summary: Let \(k\) be a positive integer divisible by 4, \(p>k\) a prime, \(f\) an elliptic cuspidal eigenform (ordinary at \(p\)) of weight \(k-1\), level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives \(ad^{0}M(-1)\) and \(ad^{0}M(2)\), where \(M\) is the motif attached to \(f\). More precisely, we prove that under certain conditions the \(p\)-adic valuation of the algebraic part of the symmetric square \(L\)-function of \(f\) evaluated at \(k\) provides a lower bound for the \(p\)-adic valuation of the order of the Pontryagin dual of the Selmer group for the adjoint of the \(p\)-adic Galois representation attached to \(f\) restricted to the Gaussian field and twisted by the inverse of the cyclotomic character. Our method uses an idea of Ribet, in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group \(U(2,2)\).
MSC:
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F80 | Galois representations |
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