an:03989449
Zbl 0612.10021
Hida, Haruzo
Galois representations into \(\text{GL}_2(\mathbb Z_p[[X]])\) attached to ordinary cusp forms
EN
Invent. Math. 85, 545-613 (1986).
00150760
1986
j
11F80 11F67 11F33 11F11 11F25 11R23
parabolic cohomology groups; Galois representations; cusp forms; Hecke algebra; Hecke operators; \(p\)-adic cusp forms; \(q\)-expansions; Iwasawa algebra; special values of the \(L\)-function
Let \(p\) be a prime \(\geq 5\), \(N\) an integer prime to \(p\), \(\Gamma\) the topological group \(1+p{\mathbb Z}_ p\), and \(\Omega\) a \(p\)-adic completion of an algebraic closure of \({\mathbb Q}_ p\). The author uses the study of the parabolic cohomology groups of \(\Gamma_ 1(Np^ r)\) to obtain results about Galois representations attached to cusp forms. Beginning with a study of the ordinary part \(h^ 0(N,{\mathbb Z}_ p)\) of the universal Hecke algebra (generated by the Hecke operators regarded as endomorphisms of the space of \(p\)-adic cusp forms of level \(N\)) the author proves the following theorem concerning \(q\)-expansions: Let \(\Lambda ={\mathbb Z}_ p[[X]]\) be the one variable Iwasawa algebra, and fix a non-trivial \(\Lambda\)-algebra homomorphism \(\lambda: h^ 0(N,{\mathbb Z})\to \Lambda\). The image under \(\lambda\) of the formal power series \(\sum^{\infty}_{n=1}T(n)q^ n\) is, when evaluated at \(X=\varepsilon (u)u^ k-1\) \((u=1+p)\), the complex \(q\)-expansion of a common eigenform \(f_{k,\varepsilon}\) for all \(T(n)\) in \(S_ k(\Gamma_ 1(p^ r))\) where \(\varepsilon: \Gamma\to \Omega^*\) is of finite order, \(\text{ker}\;\varepsilon=1+p^ r {\mathbb Z}_ p\), and \(k\) is an integer \(\geq 2\). Moreover, \(f_{k,\varepsilon}\) is not the twist by any Dirichlet character of a form of lower level.
The author then shows that one can associate to \(\lambda\) a unique Galois representation \(\pi(\lambda): \text{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \text{GL}_ 2(\Lambda)\) such that the reduction of \(\pi(\lambda)\) is equivalent to the irreducible \(\text{Gal}(\bar{\mathbb Q}/\mathbb Q)\)-representation \(\pi(f_{k,\varepsilon})\) into \(\text{GL}_ 2(\Omega)\) associated to \(f_{k,\varepsilon}\) (by Eichler-Shimura for \(f=2\), and by Deligne for \(k>2\)).
The paper ends with a study of the connections between the special values of the \(L\)-function associated to the 3-dimensional subrepresentation contained in \(\pi (f_{k,\varepsilon})\otimes {\tilde \pi}(f_{k,\varepsilon})\) (where \({\tilde \pi}\) is the contragredient representation of \(\pi\)), and the characteristic power series of a certain Iwasawa module that is associated to \(\lambda\).
S.Kamienny