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On the search of genuine \(p\)-adic modular \(L\)-functions for \(GL(n)\). With a correction to: On \(p\)-adic \(L\)-functions of \(GL(2)\times{}GL(2)\) over totally real fields. (English) Zbl 0897.11015
The study of the values of complex \(L\)-functions (Dedekind zeta function, Hasse-Weil \(L\)-function, …) at special points in one of central problems in number theory, with numerous applications to arithmetic (the Birch and Swinnerton-Dyer conjecture is one of the most important examples). To give an arithmetic interpretation, we first need to be able to interpolate (algebraic parts of) these numbers by a \(p\)-adic \(L\)-function, and then relate this function to the characteristic series of a corresponding Iwasawa module.
Kubota and Leopoldt constructed a \(p\)-adic analogue of the Riemann zeta function in 1964. Since then, a wide class of \(L\)-functions appeared admitting \(p\)-adic analogues (\(p\)-adic Hecke \(L\)-functions for totally real number fields, \(p\)-adic \(L\)-functions for CM fields, \(p\)-adic \(L\)- functions attached to Hilbert modular forms, their convolutions and symmetric squares). Taking into account these and other partial results, J. Coates and B. Perrin-Riou [Adv. Stud. Pure Math. 17, 23-54 (1989; Zbl 0783.11039)] and Dabrowski and A. Panchishkin [Proc. Symp. Pure Math. 55, P. 2, 251-292 (1994; Zbl 0837.11029)] formulated a general conjecture on the existence of bounded \(p\)-adic \(L\)-functions attached to \(p\)-admissible critical pure motives over \(\mathbb{Q}\). The admissibility criterion (or Dabrowski-Panchishkin criterion) is discussed in great detail in B. Perrin-Riou’s monograph [Astérisque 229 (1995; Zbl 0845.11040)], where also a different (conjectural) construction of \(p\)-adic \(L\)-functions of the family of Tate twists of a given motive is studied.
The present monograph contributes to the above picture with a whole array of new results and conjectures. The author states several conjectures concerning the existence and meromorphy of many variable \(p\)-adic \(L\)-functions attached to many variable irreducible Galois representations and presents supporting examples for the conjectures. He introduces the so-called genuine \(p\)-adic \(L\)-functions, which are associated to the isomorphism class of the \(p\)-adic Galois representation \(\varphi: \text{Gal} (\overline{F}/F)\to \text{GL}_n(\mathbb{I})\) (subject to a certain restriction (A1)), where \(F\) is a finite extension of \(\mathbb{Q}\) and \(\mathbb{I}\) is a normal, integral domain, finite over the completed group algebra \({\mathcal O}[[ T_n(\mathbb{Z}_p)]]\) with \({\mathcal O}\) the ring of \(p\)-adic integers in a finite extension of \(\mathbb{Q}_p\), and \(T_n\) the standard diagonal torus of \(\text{Res}_{{\mathcal O}_F/Z} \text{GL}_n\) for \({\mathcal O}_F\) which is split over \({\mathcal O}\). Such \(p\)-adic \(L\)-functions should be closely related to the characteristic ideal of the Selmer group \(\text{Sel} (\varphi^\vee)\) (defined by R. Greenberg [Proc. Symp. Pure Math. 55, P. 2, 193-223 (1994; Zbl 0819.11046)]).
Here is a summary of this monograph. Section 2 contains a summary of the theory of \(p\)-adic Hecke algebras, developed by the author in his earlier papers. General notions of congruence modules and differential modules are introduced. They are useful when describing congruences among cusp forms in terms of Hecke algebras and deformation rings of Galois representations. \(p\)-adic periods of motives under certain reducibility conditions (Red) of their local Galois representation at \(p\)-adic places are studied in Section 3. The author gives an example of vanishing \(p\)-adic periods (3.3), and he proves a general non-vanishing result for \(p\)-adic periods in (3.4). It is shown (3.5) that (Red) is equivalent to the admissibility condition when the motive is crystalline. In section 4, the author studies general \(\mathbb{I}\)-adic arithmetic Galois representations and their periods, and states conjectures on the existence of genuine \(p\)-adic \(L\)-functions (Conjecture 4.2.1, Question 4.4.1, and Conjecture 4.6.1). The main result of section 5 is Theorem 5.3.1. Here, the author expresses the periods of tensor products of rank 2 motives as monomials of periods of the components. The author studies his \(p\)-adic \(L\)-functions of \(GL(2)\times GL(2)\) (constructed by a variant of the \(p\)-adic Rankin-Selberg method [H. Hida, Ann. Inst. Fourier 41, 311-322 (1991; Zbl 0739.11019)]) in terms of the genuine \(p\)-adic \(L\)-functions of \(\varphi\otimes \rho^\vee\) for two modular Galois representations in section 6 (Theorem 6.3.2). The final two chapters are devoted to a study of the Katz \(p\)-adic \(L\)-functions interpolating Hecke \(L\)-values of CM-fields. These \(L\)- functions are close to being genuine (Theorem 8.3.1). The author establishes a version of an Artin-Tate type conjecture on the location of singularities (condition (G) in section 4.4) in this case.

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11S40 Zeta functions and \(L\)-functions
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
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