On the computation of local components of a newform. (English) Zbl 1332.11056
Math. Comput. 81, No. 278, 1179-1200 (2012); erratum ibid. 84, No. 291, 355-356 (2015).
Given a cuspidal newform \(f\) for the congruence subgroup \(\Gamma_1(N)\) of weight \(k\geq 2\) and a character \(\varepsilon\), there is a unique cuspidal automorphic representation \(\pi_f\) of \(\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})\) attached to \(f\). The Archimedean component of \(\pi_f\) is uniquely determined by the weight \(k\) of \(f\). It is the unique discrete series representation of \(\mathrm{GL}_2(\mathbb{R})\) of lowest \(\mathrm{O}(2)\)-type \(k\). The local components of \(\pi_f\) at finite primes \(p\nmid N\) are unramified representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\). They are determined uniquely by its Satake parameters, which are given in terms of \(f\) as roots of the polynomial \(X^2-a_pX+\varepsilon(p)p^{k-1}\), where \(a_p\) is the \(p\)th Fourier coefficient of \(f\).
The goal of this paper is to give an algorithm for determining the local components of \(\pi_f\) at the remaining finite primes \(p\mid N\). According to H. Carayol [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 3, 409–468 (1986; Zbl 0616.10025)], in terms of Galois representations, this problem is equivalent to the problem of determining the restriction of the local \(\lambda\)-adic Galois representations \(\rho_{f,\lambda}\) of the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) attached to \(f\) to the Galois group \(\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb{Q}_p)\) for \(p\mid N\) and primes \(\lambda\) not dividing \(p\).
If the local component \(\pi_{f,p}\) at \(p\mid N\) is not supercuspidal, the problem is not too difficult, as \(\pi_{f,p}\) is determined by the \(p\)th Fourier coefficient of \(f\) and its twists by an explicit set of Dirichlet characters of \(p\)-power conductor. If \(\pi_{f,p}\) is supercuspidal, it is induced from a type \(\tau\) of a maximal compact modulo center subgroup of \(\mathrm{GL}_2(\mathbb{Q}_p)\). Then, the strategy is to find a canonical model for \(\tau\), realized in the cohomology of a modular curve. The dual of the latter space is explicitly computed using modular symbols.
The goal of this paper is to give an algorithm for determining the local components of \(\pi_f\) at the remaining finite primes \(p\mid N\). According to H. Carayol [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 3, 409–468 (1986; Zbl 0616.10025)], in terms of Galois representations, this problem is equivalent to the problem of determining the restriction of the local \(\lambda\)-adic Galois representations \(\rho_{f,\lambda}\) of the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) attached to \(f\) to the Galois group \(\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb{Q}_p)\) for \(p\mid N\) and primes \(\lambda\) not dividing \(p\).
If the local component \(\pi_{f,p}\) at \(p\mid N\) is not supercuspidal, the problem is not too difficult, as \(\pi_{f,p}\) is determined by the \(p\)th Fourier coefficient of \(f\) and its twists by an explicit set of Dirichlet characters of \(p\)-power conductor. If \(\pi_{f,p}\) is supercuspidal, it is induced from a type \(\tau\) of a maximal compact modulo center subgroup of \(\mathrm{GL}_2(\mathbb{Q}_p)\). Then, the strategy is to find a canonical model for \(\tau\), realized in the cohomology of a modular curve. The dual of the latter space is explicitly computed using modular symbols.
Reviewer: Neven Grbac (Rijeka)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F11 | Holomorphic modular forms of integral weight |
| 11Y99 | Computational number theory |
Keywords:
cuspidal newform; cuspidal automorphic representation; family of Galois representations; local components at ramified places; explicit computationCitations:
Zbl 0616.10025References:
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