\(l\)-adic representations associated to modular forms over imaginary quadratic fields. II.

*(English)*Zbl 0823.11020This paper is part of a series which establishes an important new link between automorphic representations and arithmetic, by associating a system of \(\lambda\)-adic representations to certain modular forms over an imaginary quadratic field \(M\). [For more background, see the review of Part I: the author, D. Soudry and M. Harris, Invent. Math. 112, 377-411 (1993; Zbl 0797.11047).]

Let \(\pi\) be a cuspidal automorphic representation of \(GL(2, A_ M)\) which is regular algebraic up to twist. Thus, \(\pi\) contributes to the cohomology of the standard local system on a three manifold obtained as the quotient of hyperbolic three-space by a congruence subgroup \(GL(2, {\mathcal O}_ M)\), or equivalently \(\pi_ \infty\) has Langlands parameter \(z\mapsto \left( \begin{smallmatrix} z^{1-k} &0\\ 0 &\overline {z}^{k-1} \end{smallmatrix} \right)\), with \(k\geq 2\) an integer. Suppose further that the central character of \(\pi\) is Galois-invariant, and \(k\) is even. For good places \(v\), let \(\{ \alpha_ v, \beta_ v\}\) denote the Langlands parameters of \(\pi_ v\), and let \(F_ \pi\) denote the field generated by the \(\alpha_ v+ \beta_ v\) and \(\alpha_ v \beta_ v\) for all good places \(v\); \(F_ \pi\) is a number field.

Then the main theorem of the author states that there is an extension \(E/ F_ \pi\) of degree at most four and for each prime \(\lambda\) of \(F_ \pi\) there is a continuous irreducible representation \(\rho\) of \(\text{Gal} (\overline {M}/ M)\) to \(GL_ 2 (E_{\lambda'} )\) (\(\lambda'\) a prime of \(E\) over \(\lambda\)) such that on the complement of a set of Dirichlet density zero the characteristic polynomial of \(\rho (\text{Frob}_ v)\) is \((X- \alpha_ v) (X-\beta_ v)\).

The proof of this result uses the work of Harris, Soudry, and the author (loc. cit.) to associate to \(\pi\) and its twists a family of holomorphic Siegel modular forms of genus 2. The forms obtained are of low weight and do not contribute to the \(l\)-adic cohomology of sheaves on Siegel three- folds. However, using the results of the author [Duke Math. J. 63, 281- 332 (1991; Zbl 0810.11033); On the \(l\)-adic cohomology of Siegel threefolds, preprint, per biblio.] one may attach \(l\)-adic representations to them. The difficulty is that one does not know the trace of the Frobenius in general, but rather degree four polynomials satisfied by the image of Frobenius elements. Using this information, the author analyzes the Zariski closure of the image of these representations, and obtains his results. Among other ingredients, he needs a refinement of the strong multiplicity one theorem, established in an appendix by D. Ramakrishnan (see the following review).

In the last section the author combines his work with the Faltings-Serre method to check that for certain \(\pi\) corresponding to elliptic curves over imaginary quadratic fields the dual of the Tate module is isomorphic to the \(l\)-adic representation constructed. This allows him to check the Ramanujan conjecture for such \(\pi\), and to prove results about the \(L\)- function of the corresponding elliptic curve.

Let \(\pi\) be a cuspidal automorphic representation of \(GL(2, A_ M)\) which is regular algebraic up to twist. Thus, \(\pi\) contributes to the cohomology of the standard local system on a three manifold obtained as the quotient of hyperbolic three-space by a congruence subgroup \(GL(2, {\mathcal O}_ M)\), or equivalently \(\pi_ \infty\) has Langlands parameter \(z\mapsto \left( \begin{smallmatrix} z^{1-k} &0\\ 0 &\overline {z}^{k-1} \end{smallmatrix} \right)\), with \(k\geq 2\) an integer. Suppose further that the central character of \(\pi\) is Galois-invariant, and \(k\) is even. For good places \(v\), let \(\{ \alpha_ v, \beta_ v\}\) denote the Langlands parameters of \(\pi_ v\), and let \(F_ \pi\) denote the field generated by the \(\alpha_ v+ \beta_ v\) and \(\alpha_ v \beta_ v\) for all good places \(v\); \(F_ \pi\) is a number field.

Then the main theorem of the author states that there is an extension \(E/ F_ \pi\) of degree at most four and for each prime \(\lambda\) of \(F_ \pi\) there is a continuous irreducible representation \(\rho\) of \(\text{Gal} (\overline {M}/ M)\) to \(GL_ 2 (E_{\lambda'} )\) (\(\lambda'\) a prime of \(E\) over \(\lambda\)) such that on the complement of a set of Dirichlet density zero the characteristic polynomial of \(\rho (\text{Frob}_ v)\) is \((X- \alpha_ v) (X-\beta_ v)\).

The proof of this result uses the work of Harris, Soudry, and the author (loc. cit.) to associate to \(\pi\) and its twists a family of holomorphic Siegel modular forms of genus 2. The forms obtained are of low weight and do not contribute to the \(l\)-adic cohomology of sheaves on Siegel three- folds. However, using the results of the author [Duke Math. J. 63, 281- 332 (1991; Zbl 0810.11033); On the \(l\)-adic cohomology of Siegel threefolds, preprint, per biblio.] one may attach \(l\)-adic representations to them. The difficulty is that one does not know the trace of the Frobenius in general, but rather degree four polynomials satisfied by the image of Frobenius elements. Using this information, the author analyzes the Zariski closure of the image of these representations, and obtains his results. Among other ingredients, he needs a refinement of the strong multiplicity one theorem, established in an appendix by D. Ramakrishnan (see the following review).

In the last section the author combines his work with the Faltings-Serre method to check that for certain \(\pi\) corresponding to elliptic curves over imaginary quadratic fields the dual of the Tate module is isomorphic to the \(l\)-adic representation constructed. This allows him to check the Ramanujan conjecture for such \(\pi\), and to prove results about the \(L\)- function of the corresponding elliptic curve.

Reviewer: S.Friedberg (Santa Cruz)

##### MSC:

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F33 | Congruences for modular and \(p\)-adic modular forms |

11F75 | Cohomology of arithmetic groups |

11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |

11F55 | Other groups and their modular and automorphic forms (several variables) |

11G05 | Elliptic curves over global fields |

11F80 | Galois representations |

14G20 | Local ground fields in algebraic geometry |

##### Keywords:

cohomology of hyperbolic three-manifolds; automorphic representations; holomorphic Siegel modular forms; \(l\)-adic representations; elliptic curves over imaginary quadratic fields; Tate module; Ramanujan conjecture; \(L\)-function##### References:

[1] | [BFH] Bump, D., Friedberg, S., Hoffstein, G.: Private communication from S. Friedberg |

[2] | [Cl] Clozel, L.: Motifs et Formes Automorphes: Applications du Pricipe de Fonctorialité. In: Clozel, L., Milne, J.S. (eds.) Automorphic Forms, Shimua Varieties andL-functions, vol. 1. New York: Academic Press 1990 · Zbl 0705.11029 |

[3] | [Cr] Cremona, J.: Modular Symbols for Quadratic Fields. Compos. Math.51, 275-323 (1984) · Zbl 0546.14027 |

[4] | [EGM] Elstrodt, J., Grunewald, F., Menicke, J.: On the GroupPSL 2(?[i]). In: Armitage, J.V. (ed.) Journées Arithmétiques. (Lond. Math. Soc. Lect. Notes Ser., vol. 56) Cambridge: Cambridge University Press 1982 |

[5] | [GL] Gérardin, P., Labesse, J.-P.: The Solution of a Base Change Problem for GL(2) (Following Langlands, Saito and Shintani). In: Borel, A., Casselman, W. (eds.) Automorphic Forms, Representations andL-functions. (Proc. Symp. Pure Math., vol. XXXIII, part 2) Providence, RI: Am. Math. Soc. 1979 |

[6] | [Ha] Harder, G.: Eisenstein Cohomology of Arithmetic Groups. The Case GL2. Invent. Math.89, 37-118 (1987) · Zbl 0629.10023 · doi:10.1007/BF01404673 |

[7] | [HST] Harris, M., Soudry, D., Taylor, R.:l-adic Representations Associated to Modular Forms over Imaginary Quadratic Fields. I. Lifting toGSp 4(?). Invent. math.112, 377-411 (1993) · Zbl 0797.11047 · doi:10.1007/BF01232440 |

[8] | [He] Henniart, G.: Représentationsl-adiques Abéliennes. In: Bertin, M.-J. (ed.) Séminaires de Théorie des Nombres, Paris 1980-81, pp. 107-126. Boston Basel Stuttgart: Birkhäuser 1982 |

[9] | [JS] Jacquet, H., Shalika, J.: On Euler Products and the Classification of Automorphic Representations. I. Am. J. Math.103(2), 499-558 (1981) · Zbl 0473.12008 · doi:10.2307/2374103 |

[10] | [L] Livné, R.: Cubic Exponential Sums and Galois Representations. Contemp. Math.67, 247-261 (1987) · Zbl 0621.14019 |

[11] | [R] Ramakrishnan, D.: A Refinement of the Strong Multiplicity One Theorem for GL(2). Appendix to this paper. Invent. math.116, 619-643 (1994) · Zbl 0823.11021 · doi:10.1007/BF01231576 |

[12] | [Se] Serre, J.-P.: Abelianl-adic Representations and Elliptic Curves. New York: Benjamin 1968 |

[13] | [Sh] Shahidi, F.: On CertainL-functions. Am. J. Math.103(2), 297-355 (1981) · Zbl 0467.12013 · doi:10.2307/2374219 |

[14] | [T1] Taylor, R.: Galois Representations Associated to Siegel Modular Forms of Low Weight. Duke Math. J.63 (1991) · Zbl 0810.11033 |

[15] | [T2] Taylor, R.: On thel-adic Cohomology of Siegel Three Folds. (Preprint) |

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