\(p\)-adic \(L\)-functions and \(p\)-adic representations.
(Fonctions \(L\) \(p\)-adiques des représentations \(p\)-adiques.)

*(French)*Zbl 0845.11040
Astérisque. 229. Paris: Société Mathématique de France, 198 p. (1995).

The original construction of \(p\)-adic \(L\)-functions was analytic and was based on a \(p\)-adic interpolation of values of Dirichlet \(L\)-functions. The main conjecture of Iwasawa theory, proved by B. Mazur and A. Wiles [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)], showed that these \(p\)-adic \(L\)-functions could also be obtained as characteristic power series of certain algebraically defined Iwasawa modules, for example the inverse limit of the ideal class groups in a cyclotomic \(\mathbb{Z}_p\)-extension. A third approach, which is somewhat between the analytic and algebraic approaches, had its origins in a paper of K. Iwasawa [J. Math. Soc. Japan 16, 42-82 (1964; Zbl 0125.29207)]. It produces \(p\)-adic \(L\)-functions as characteristic power series of modules defined in terms of local units modulo cyclotomic units. This approach was refined by R. Coleman [Invent. Math. 53, 91-116 (1979; Zbl 0429.12010)] and proved useful in the work of J. Coates and A. Wiles [J. Aust. Math. Soc., Ser. A 26, 1-25 (1978; Zbl 0442.12007)] on the conjecture of Birch and Swinnerton-Dyer.

More recently, \(p\)-adic \(L\)-functions have been constructed in various other cases and main conjectures have been formulated, for example for modular or CM elliptic curves having good ordinary reduction at \(p\). In the case of supersingular reduction, two \(p\)-adic \(L\)-functions could be constructed via \(p\)-adic interpolation. Since these functions do not arise from the Iwasawa algebra and since the natural choice for an Iwasawa module does not have torsion, the formulation of a main conjecture in this case presented difficulties. These were overcome by an analogue of the third approach in work of the author [Ann. Inst. Fourier (Grenoble) 43, No. 4, 945-995 (1993; Zbl 0840.11024)]. The purpose of the present book is to explain this approach in a much more general setting. Before giving a more detailed description, we need some notation. Let \(p\) be an odd prime. Let \(F/\mathbb{Q}\) be a finite extension unramified at \(p\) and let \(S\) be a finite set of primes of \(F\) containing the infinite primes and the primes above \(p\). Let \(V\) be a representation of \(\text{Gal}(\overline F/F)\) unramified outside \(S\) such that \(V\) is crystalline at all primes dividing \(p\). Let \(V^*(1)= \text{Hom}_{\mathbb{Q}_p}(V, \mathbb{Q}_p(1))\). Let \(D_p(V)\) be the crystalline Dieudonné module attached to \(\text{Ind}_{F/\mathbb{Q}}(V)\) and similarly for \(V^*(1)\). Let \(F_n= F(\mu_{p^{n 1}})\) and \(G_n= \text{Gal}(F_n/ F)\). Then \({\mathcal H}(G_{\infty})\) can be loosely described as those elements of \(\mathbb{Q}_p[[G_{\infty}]]\) satisfying certain growth conditions. In particular, it contains the Iwasawa algebra \(\mathbb{Z}_p[[G_{\infty}]]\). Let \(\mathbb{K}\) be the total ring of fractions of \(B_{\text{cris}}\otimes_{\mathbb{Q}_p}{\mathcal H}(G_{\infty})\).

In the first two chapters, the author starts with a lattice \({\mathbf T}\subset V\) and defines a \(\Lambda\)-submodule \(\mathbb{I}_{\text{arith},\{p\}}({\mathbf T})\), called the module of \(p\)-adic \(L\)-functions of \({\mathbf T}\), lying inside a certain exterior power of \(\mathbb{K}\otimes D_p(V^*(1))\). This is first defined without the factors at infinity, then refined to include these missing factors. Under the assumption of certain hypotheses similar to the weak Leopoldt conjecture, this module is free of rank one over \(\Lambda\). There is a functional equation relating \(\mathbb{I}_{\text{arith}, \{p\}}({\mathbf T})\) and \(\mathbb{I}_{\text{arith},\{p\}}({\mathbf T}^*(1))\).

The main tool in the construction of \(\mathbb{I}_{\text{arith},\{p\}}({\mathbf T})\) is a family of \(\mathbb{K}\)-module homomorphisms \[ \Omega_{V, h, \mathbb{K}}^{\epsilon}: D_p(V)\otimes \mathbb{K}\to \underset\leftarrow\lim \oplus_{v|p} H^1(F_{n, v}, {\mathbf T})\otimes \mathbb{K}. \] These maps were used by the author in [Invent. Math. 115, No. 1, 81-149 (1994; Zbl 0838.11071)] and generalize the inverse of a homomorphism constructed by Coleman, where he assigns a power series to a norm compatible sequence of local units.

In the third chapter, the order of vanishing of \(\mathbb{I}_{\text{arith}, \{p\}}({\mathbf T})\) at the trivial character is determined under a suitable regularity hypothesis, and the “leading term” is evaluated in terms of \(p\)-adic periods.

In the final chapter, the author starts with a motivic structure over \(\mathbb{Q}\) whose \(p\)-adic realization is crystalline. Using the work of the third chapter, she conjectures the existence of a distinguished generator of \(\mathbb{I}_{\text{arith}, \{p\}}({\mathbf T})\) and relates it to the analytic and algebraic (conjecturally defined) \(p\)-adic \(L\)-functions.

The book concludes with appendices on Galois cohomology related to Iwasawa theory, on the weak Leopoldt conjecture, and on local Tamagawa numbers. Overall, this book makes a substantial contribution to the construction and study of new \(p\)-adic \(L\)-functions. It should have an influence on research in this area for many years.

More recently, \(p\)-adic \(L\)-functions have been constructed in various other cases and main conjectures have been formulated, for example for modular or CM elliptic curves having good ordinary reduction at \(p\). In the case of supersingular reduction, two \(p\)-adic \(L\)-functions could be constructed via \(p\)-adic interpolation. Since these functions do not arise from the Iwasawa algebra and since the natural choice for an Iwasawa module does not have torsion, the formulation of a main conjecture in this case presented difficulties. These were overcome by an analogue of the third approach in work of the author [Ann. Inst. Fourier (Grenoble) 43, No. 4, 945-995 (1993; Zbl 0840.11024)]. The purpose of the present book is to explain this approach in a much more general setting. Before giving a more detailed description, we need some notation. Let \(p\) be an odd prime. Let \(F/\mathbb{Q}\) be a finite extension unramified at \(p\) and let \(S\) be a finite set of primes of \(F\) containing the infinite primes and the primes above \(p\). Let \(V\) be a representation of \(\text{Gal}(\overline F/F)\) unramified outside \(S\) such that \(V\) is crystalline at all primes dividing \(p\). Let \(V^*(1)= \text{Hom}_{\mathbb{Q}_p}(V, \mathbb{Q}_p(1))\). Let \(D_p(V)\) be the crystalline Dieudonné module attached to \(\text{Ind}_{F/\mathbb{Q}}(V)\) and similarly for \(V^*(1)\). Let \(F_n= F(\mu_{p^{n 1}})\) and \(G_n= \text{Gal}(F_n/ F)\). Then \({\mathcal H}(G_{\infty})\) can be loosely described as those elements of \(\mathbb{Q}_p[[G_{\infty}]]\) satisfying certain growth conditions. In particular, it contains the Iwasawa algebra \(\mathbb{Z}_p[[G_{\infty}]]\). Let \(\mathbb{K}\) be the total ring of fractions of \(B_{\text{cris}}\otimes_{\mathbb{Q}_p}{\mathcal H}(G_{\infty})\).

In the first two chapters, the author starts with a lattice \({\mathbf T}\subset V\) and defines a \(\Lambda\)-submodule \(\mathbb{I}_{\text{arith},\{p\}}({\mathbf T})\), called the module of \(p\)-adic \(L\)-functions of \({\mathbf T}\), lying inside a certain exterior power of \(\mathbb{K}\otimes D_p(V^*(1))\). This is first defined without the factors at infinity, then refined to include these missing factors. Under the assumption of certain hypotheses similar to the weak Leopoldt conjecture, this module is free of rank one over \(\Lambda\). There is a functional equation relating \(\mathbb{I}_{\text{arith}, \{p\}}({\mathbf T})\) and \(\mathbb{I}_{\text{arith},\{p\}}({\mathbf T}^*(1))\).

The main tool in the construction of \(\mathbb{I}_{\text{arith},\{p\}}({\mathbf T})\) is a family of \(\mathbb{K}\)-module homomorphisms \[ \Omega_{V, h, \mathbb{K}}^{\epsilon}: D_p(V)\otimes \mathbb{K}\to \underset\leftarrow\lim \oplus_{v|p} H^1(F_{n, v}, {\mathbf T})\otimes \mathbb{K}. \] These maps were used by the author in [Invent. Math. 115, No. 1, 81-149 (1994; Zbl 0838.11071)] and generalize the inverse of a homomorphism constructed by Coleman, where he assigns a power series to a norm compatible sequence of local units.

In the third chapter, the order of vanishing of \(\mathbb{I}_{\text{arith}, \{p\}}({\mathbf T})\) at the trivial character is determined under a suitable regularity hypothesis, and the “leading term” is evaluated in terms of \(p\)-adic periods.

In the final chapter, the author starts with a motivic structure over \(\mathbb{Q}\) whose \(p\)-adic realization is crystalline. Using the work of the third chapter, she conjectures the existence of a distinguished generator of \(\mathbb{I}_{\text{arith}, \{p\}}({\mathbf T})\) and relates it to the analytic and algebraic (conjecturally defined) \(p\)-adic \(L\)-functions.

The book concludes with appendices on Galois cohomology related to Iwasawa theory, on the weak Leopoldt conjecture, and on local Tamagawa numbers. Overall, this book makes a substantial contribution to the construction and study of new \(p\)-adic \(L\)-functions. It should have an influence on research in this area for many years.

Reviewer: L.Washington (College Park)

##### MSC:

11R42 | Zeta functions and \(L\)-functions of number fields |

11R23 | Iwasawa theory |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |