Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts. (English) Zbl 1439.11119
Author’s abstract: In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form \(f\) on the unitary group \(U(n, n)(\mathbb A_F)\) for a large class of totally real fields \(F\) via a divisibility of a special value of the standard \(L\)-function associated to \(f\). We also study \(l\)-adic properties of the Fourier coefficients of an Ikeda lift \(I_\phi\) (of an elliptic modular form \(\phi\)) on \(U(n, n)(\mathbb A_{\mathbb Q})\), proving that they are \(l\)-adic integers which do not all vanish modulo \(l\). Finally we combine these results to show that the condition of \(l\) being a congruence prime for \(I_\phi\) is controlled by the \(l\)-divisibility of a product of special values of the symmetric square \(L\)-function of \(\phi\). We close the paper by computing an example when our main theorem applies.
Reviewer: Goran Muić (Zagreb)
MSC:
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F32 | Modular correspondences, etc. |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
Software:
SageMathReferences:
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