Companion forms for unitary and symplectic groups. (English) Zbl 1295.11043

Gee, Toby; Geraghty, David

Duke Math. J. 161, No. 2, 247-303 (2012).

From the text: “The problem of companion forms was first introduced by J.-P. Serre [ibid. J. 54, 179–230 (1987; Zbl 0641.10026)] for modular forms in his seminal paper. Fix a prime \(l\), algebraic closures \(\overline{\mathbb Q}\) and \(\overline{\mathbb Q_l}\) of \(\mathbb Q\) and \(\mathbb Q_l\), respectively, and an embedding of \(\overline{\mathbb Q}\) into \(\overline{\mathbb Q_l}\). Suppose that \(f\) is a modular newform of weight \(k> 2\) which is ordinary at \(l\), so that the corresponding \(l\)-adic Galois representation \(\rho_{f,l}\) becomes reducible when restricted to a decomposition group \(G_{\mathbb Q_l}\) at \(l\). Then the companion forms problem is essentially the question of determining for which other weights \(k'\) there is an ordinary newform \(g\) of weight \(k'\geq 2\) such that the Galois representations \(\rho_{f,l}\) and \(\rho_{g,l}\) are congruent modulo \(l\). The problem is straightforward unless the restriction to \(G_{\mathbb Q_l}\) of \(\overline\rho_{f,l}\) (the reduction mod \(l\) of \(\rho_{f,l}\)) is split and nonscalar, in which case there are two possible Hida families whose corresponding Galois representations lift \(\overline\rho_{f,l}\). The restrictions of the corresponding Galois representations to a decomposition group at \(l\) are either “upper triangular” or “lower triangular”.”
“This problem was essentially resolved by B. H. Gross [ibid. 61, No. 2, 445–517 (1990; Zbl 0743.11030)] and R. F. Coleman and J. F. Voloch [Invent. Math. 110, No. 2, 263–281 (1992; Zbl 0770.11024)]. In the paper [ibid. 136, No. 2, 275–320 (2007; Zbl 1121.11039)], the first author reproved these results and generalized them to Hilbert modular forms, by a completely new technique.”
“We prove a companion forms theorem for ordinary \(n\)-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author and a technique for deducing companion forms theorems developed by the first author. We deduce results about the possible Serre weights of \(\mod l\) Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of \(\mathrm{GSp}_4\) over totally real fields.”

Reviewer: Andrzej Dąbrowski (Szczecin)

MSC:

11F33	Congruences for modular and \(p\)-adic modular forms
11F55	Other groups and their modular and automorphic forms (several variables)
11F80	Galois representations

Keywords:

comparison forms; Galois representation; automorphic representation; Serre weights; universal deformation ring; automorphy lifting; functoriality principle

Citations:

Zbl 0641.10026; Zbl 0743.11030; Zbl 0770.11024; Zbl 1121.11039

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