\(p\)-adic families of automorphic forms for \(\text{GL}_n\).
(Familles \(p\)-adiques de formes automorphes pour \(\text{GL}_n\).)

*(French)*Zbl 1093.11036Fix a prime number \(p\) and let \(\mathbb C_p\) denote the completion of an algebraic closure of the field \(\mathbb Q_p\) of \(p\)-adic numbers. Let \(N\) be an integer relatively prime to \(p\).

By the work of Hida we know that any eigenform \(f\) on \(\Gamma_0(pN)\) of weight \(k\geq 2\), with Fourier coefficients in \(\mathbb C_p\) and slope 0, is a member of a \(p\)-adic analytic family \(f_\kappa\) of overconvergent \(p\)-adic modular eigenforms of slope 0 parametrized by \(p\)-adic weights \(\kappa\) for \(\kappa\) ranging through a small \(p\)-adic neighborhood of \(k\). R. Coleman and B. Mazur [In: “Galois representations in arithmetic algebraic geometry” (ed. by A. J. Scholl and R. L. Taylor), Cambridge Univ. Press, 1–113 (1998; Zbl 0932.11030)], constructed “eigencurves”, rigid analytic curves over \(\mathbb Q_p\) whose \(\mathbb C_p\)-valued points parametrise all finite slope normalized overconvergent \(p\)-adic modular eigenforms with Fourier coefficients in \(\mathbb C_p\). K. Buzzard [In: “Modular curves and abelian varieties” (ed. by J. Cremona et al.), Prog. Math. 224, 23–44 (2004; Zbl 1166.11322)] formulated definitions of overconvergent \(p\)-adic automorphic forms for \(\text{GL}_1\) over a number field, and for \(D^\times\), \(D\) a definite quaternion algebra over \(\mathbb Q\).

The author gives definitions for \(p\)-adic automorphic forms on any twisted form \(G\) of \(\text{GL}_n/\mathbb Q\) compact at infinity, and constructs the “eigenvariety” of finite slope eigenforms of wild level \(\Gamma_0(p)\), at a split place \(p\). He proves variants of Coleman’s result that small slope forms are classical (Prop. 4.7.4) and of Wan’s bounds for explicit radii for the families (Cor. 5.3.3). The eigenvariety should perhaps be the natural domain for the special values of \(p\)-adic \(L\)-functions.

Let us briefly describe the contents of each section.

Sections 2 and 3 summarize necessary results concerning representation theory of \(\text{GL}_n(\mathbb Q_p)\) and its Iwahori subgroup \(\Gamma_0(p)\). Let \(U_0(p)\subset G(\mathbb A_f)\) denote a compact open subgroup whose \(p\)-component is \(\Gamma_0(p)\).

In section 4 the author defines the analytic family \({\mathcal S}(G,U_0(p))=\{{\mathcal S}_t(G,U_0 (p))\}_{t\in{\mathcal W}(\mathbb C_p)}\) of Banach spaces of \(p\)-adic automorphic forms of type \((G,U_0(p))\) and weights in \({\mathcal W} (\mathbb C_p): =\operatorname{Hom}_{gr-an} ((\mathbb Z_p^\times)^n, \mathbb C^\times_p)\). Such a family admits an action of the global Hecke algebra. In section 5 he studies characteristic series of the \(U_p\)-operator acting on \({\mathcal S}(G,U_0 (p))\). The main construction (generalizing the construction of Coleman-Mazur) is given in section 6 (Thm. 6.3.6, Prop. 6.4.2, Prop. 6.4.6). As an application, in section 7 he constructs \(n\)-dimensional \(p\)-adic families of non-ordinary, \(n\)-dimensional Galois representations coming from Shimura varieties of certain unitary groups.

By the work of Hida we know that any eigenform \(f\) on \(\Gamma_0(pN)\) of weight \(k\geq 2\), with Fourier coefficients in \(\mathbb C_p\) and slope 0, is a member of a \(p\)-adic analytic family \(f_\kappa\) of overconvergent \(p\)-adic modular eigenforms of slope 0 parametrized by \(p\)-adic weights \(\kappa\) for \(\kappa\) ranging through a small \(p\)-adic neighborhood of \(k\). R. Coleman and B. Mazur [In: “Galois representations in arithmetic algebraic geometry” (ed. by A. J. Scholl and R. L. Taylor), Cambridge Univ. Press, 1–113 (1998; Zbl 0932.11030)], constructed “eigencurves”, rigid analytic curves over \(\mathbb Q_p\) whose \(\mathbb C_p\)-valued points parametrise all finite slope normalized overconvergent \(p\)-adic modular eigenforms with Fourier coefficients in \(\mathbb C_p\). K. Buzzard [In: “Modular curves and abelian varieties” (ed. by J. Cremona et al.), Prog. Math. 224, 23–44 (2004; Zbl 1166.11322)] formulated definitions of overconvergent \(p\)-adic automorphic forms for \(\text{GL}_1\) over a number field, and for \(D^\times\), \(D\) a definite quaternion algebra over \(\mathbb Q\).

The author gives definitions for \(p\)-adic automorphic forms on any twisted form \(G\) of \(\text{GL}_n/\mathbb Q\) compact at infinity, and constructs the “eigenvariety” of finite slope eigenforms of wild level \(\Gamma_0(p)\), at a split place \(p\). He proves variants of Coleman’s result that small slope forms are classical (Prop. 4.7.4) and of Wan’s bounds for explicit radii for the families (Cor. 5.3.3). The eigenvariety should perhaps be the natural domain for the special values of \(p\)-adic \(L\)-functions.

Let us briefly describe the contents of each section.

Sections 2 and 3 summarize necessary results concerning representation theory of \(\text{GL}_n(\mathbb Q_p)\) and its Iwahori subgroup \(\Gamma_0(p)\). Let \(U_0(p)\subset G(\mathbb A_f)\) denote a compact open subgroup whose \(p\)-component is \(\Gamma_0(p)\).

In section 4 the author defines the analytic family \({\mathcal S}(G,U_0(p))=\{{\mathcal S}_t(G,U_0 (p))\}_{t\in{\mathcal W}(\mathbb C_p)}\) of Banach spaces of \(p\)-adic automorphic forms of type \((G,U_0(p))\) and weights in \({\mathcal W} (\mathbb C_p): =\operatorname{Hom}_{gr-an} ((\mathbb Z_p^\times)^n, \mathbb C^\times_p)\). Such a family admits an action of the global Hecke algebra. In section 5 he studies characteristic series of the \(U_p\)-operator acting on \({\mathcal S}(G,U_0 (p))\). The main construction (generalizing the construction of Coleman-Mazur) is given in section 6 (Thm. 6.3.6, Prop. 6.4.2, Prop. 6.4.6). As an application, in section 7 he constructs \(n\)-dimensional \(p\)-adic families of non-ordinary, \(n\)-dimensional Galois representations coming from Shimura varieties of certain unitary groups.

Reviewer: Andrzej Dąbrowski (Szczecin)

##### MSC:

11F85 | \(p\)-adic theory, local fields |

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

11G18 | Arithmetic aspects of modular and Shimura varieties |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |