# zbMATH — the first resource for mathematics

Relations monomiales entre périodes p-adiques. (Monomial relations between p-adic periods). (French) Zbl 0658.14023
Let K be a number field which is an imaginary quadratic extension of a totally real field and consider the family S of complex abelian varieties with complex multiplication by the ring of integers in K. By a theorem of Shimura, S is defined over a number field. One can define a canonical differential m-form $$\Omega$$ ($$\tau)$$ on S (depending on an embedding $$\tau:\quad K\hookrightarrow {\mathbb{C}}),$$ and comparing with a suitable algebraic m-form $$\Omega^{alg}(\tau)$$ on S one obtains a complex period P($$\tau)$$ on S which is constant as $$\Omega$$ and $$\Omega^{alg}$$ are horizontal for the Gauß-Manin connection.
On the other hand one can define periods p($$\tau$$,$$\Phi)$$ for each CM- type $$\Phi$$ by suitable integration. Comparing P($$\tau)$$ and p($$\tau$$,$$\Phi)$$ the author obtains a new proof of a theorem of Shimura saying that for two families $$(\Phi_ i)$$, $$(\Phi '_ i)$$, $$i=1,...,m$$ of CM-types with equal formal sum the products of the periods are equal up to an algebraic number. The main achievement of the paper is an analogue construction of p-adic periods (for a prime p at which the given abelian variety has ordinary good reduction) and to prove the analogue of Shimura’s theorem for these p-adic periods, thus confirming a conjecture of de Shalit.
Reviewer: F.Herrlich

##### MSC:
 14K22 Complex multiplication and abelian varieties 14G25 Global ground fields in algebraic geometry 32G20 Period matrices, variation of Hodge structure; degenerations 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 11R80 Totally real fields
Full Text:
##### References:
 [1] [B] Blasius, D.: Travail en cours de r?daction. M.S.R.I. 1987 [2] [BBM] Berthelot, P., Breen, L., Messing, W.: Th?orie de Dieudonn? cristalline II. (Lecture Notes in Math., Vol. 930) Berlin-Heidelberg-New-York: Springer 1982 · Zbl 0516.14015 [3] [C] Carayol, H.: Sur la mauvaise r?duction des courbes de Shimura. Compos. Math.59, 151-230 (1986) · Zbl 0607.14021 [4] [D1] Deligne, P.: Equations diff?rentielles ? points singuliers r?guliers. (Lecture Notes in Math., Vol. 163). Berlin-Heidelberg-New-York: Springer 1970 [5] [D2] Deligne, P.: Travaux de Shimura. S?m. Bourbaki, 389, 1971 [6] [D3] Deligne, P.: Hodge Cycles on abelian varieties, Hodge cycles, motives and Shimura varieties (r?dig? par J.S. Milne). (Lecture Notes in Math., Vol. 900, pp. 9-101). Berlin-Heidelberg-New-York: Springer 1972 [7] [DR] Deligne, P., Rapoport, M.: Les Sch?mas de modules des courbes elliptiques, Modular functions of one variable II. (Lecture Notes in Math., Vol. 349). Berlin-Heidelberg-New-York: Springer 1973 [8] [DS] De Shalit, E.: On Katz’zp-adicL functions for CM fields and coherent units. Preprint 1985 [9] [DS1] De Shalit, E.: Monomial relations betweenp-adic periods. J. Reine Angew. Math.374, 193-207 (1987) · Zbl 0597.14038 [10] [EGA] Grothendieck, A., Dieudonn?, J.: El?ments de g?om?trie alg?brique. (Grundlehren Math. W., Bd. 166). Berlin-Heidelberg-New-York: Springer 1971 [11] [Fa1] Faltings, G.: Arithmetische Kompaktifizierung des Modulsraums des abelschen varietaten. Arbeitstagung Bonn 1984. (Lecture Notes in Math., Vol. 1111, pp. 318-383). Berlin-Heidelberg-New-York: Springer 1985 [12] [Fa2] Faltings, G.:p-adic Hodge Theory. J. Am. Math Soc.1, 255-299 (1988) · Zbl 0764.14012 [13] [Gi] Gillard, R.: Etude d’une famille modulaire de vari?t?s ab?liennes. (Preprint Genoble, 1987) [14] [Gr] Gross, B.: On the periods of abelian integrals and a formula of Chowla and Selberg. Invent. Math.45, 193-211 (1978) · Zbl 0418.14023 [15] [J] Jacobowitz, R.: Hermitian forms over local fields. Am. J. Math.84, 441-465 (1962) · Zbl 0118.01901 [16] [K1] Katz, N.: Travaux de Dwork. S?minaire Bourbaki 409, 1972 [17] [K2] Katz, N.: Serre-Tate moduli, Surfaces alg?briques. (Lecture Notes in Math., Vol. 868, pp. 138-202). Berlin-Heidelberg-New-York: Springer 1978 [18] [K3] Katz, N.:p-adicL functions for CM fields. Invent. Math.49, 199-297 (1978) · Zbl 0417.12003 [19] [KM] Katz, N., Mazur, B.: Arithmetic moduli of elliptic curves. Ann. Math. Stud.108 (1985) · Zbl 0576.14026 [20] [Ko] Koblitz, N.:p-adic variation of the zeta function over families of varieties defined over finite fields. Compos. Math.31, 119-218 (1975) · Zbl 0332.14008 [21] [KO] Katz, N., Oda, T.: On the differentiation of De Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ.8, 199-213 (1968) · Zbl 0165.54802 [22] [L] Lang, S.: Abelian varieties. Berlin-Heidelberg-New-York: Springer 1983 · Zbl 0516.14031 [23] [Me] Messing, W.: The crystals associated to Barsotti-Tate groups: with applications to abelian schemes (Lecture Nobes in Math., Vol. 264). Berlin-Heidelberg-New-York: Springer 1972 · Zbl 0243.14013 [24] [Mi] Miyake, K.: Models of certain automorphic function fields. Acta Math.126, 245-307 (1971) · Zbl 0212.53203 [25] [Mu] Mumford, D.: Geometric invariant theory. (Ergeb. Math., Vol. 34). Berlin-Heidelberg-New-York: Springer 1965 · Zbl 0147.39304 [26] [Sc] Schappacher, N.: On the periods of Hecke characters. MPI 86-4, Bonn, 1986 [27] [S1] Shimura, G.: On analytical families of polarized abelian varieties and automorphic functions. Ann. Math.78, 149-192 (1963) · Zbl 0142.05402 [28] [S2] Shimura, G.: Arithmetic of unitary groups. Ann. Math.79, 369-409 (1964) · Zbl 0144.29504 [29] [S3] Shimura, G.: On the field of definition for a field of automorphic functions, III. Ann. Math.83, 377-385 (1966) · Zbl 0222.14027 [30] [S4] Shimura, G.: Automorphic forms and the periods of abelian varieties. J. Math. Soc. Japan31, 561-592 (1979) · Zbl 0456.10015 [31] [S5] Shimura, G.: The arithmetric of certain zeta functions and automorphic forms on orthogonal groups. Ann. Math.111, 313-375 (1980) · Zbl 0438.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.