Invariants and coinvariants of semilocal units modulo elliptic units
Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 487-504.

Soient p un nombre premier, et k un corps quadratique imaginaire dans lequel p se décompose en deux idéaux maximaux 𝔭 et 𝔭 ¯. Soit k l’unique p -extension de k non ramifiée en dehors de 𝔭, et soit K une extension finie de k , abélienne sur k. Soit 𝒰 /𝒞 la limite projective du module des unités semi-locales principales modulo le module des unités elliptiques. Nous prouvons que les différents modules des invariants et des co-invariants de 𝒰 /𝒞 sont finis. Notre approche utilise les distributions et la fonction L p-adique, définie dans [5].

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two primes 𝔭 and 𝔭 ¯. Let k be the unique p -extension of k which is unramified outside of 𝔭, and let K be a finite extension of k , abelian over k. Let 𝒰 /𝒞 be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of 𝒰 /𝒞 are finite. Our approach uses distributions and the p-adic L-function, as defined in [5].

DOI : 10.5802/jtnb.808
Viguié, Stéphane 1

1 Université de Franche-Comté 16 route de Gray 25030 Besançon cedex, France
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Viguié, Stéphane. Invariants and coinvariants of semilocal units modulo elliptic units. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 487-504. doi : 10.5802/jtnb.808. http://www.numdam.org/articles/10.5802/jtnb.808/

[1] J-R. Belliard, Global Units modulo Circular Units: descent without Iwasawa’s Main Conjecture. Canadian J. Math. 61 (2009), 518–533. | MR

[2] W. Bley, On the Equivariant Tamagawa Number Conjecture for Abelian Extensions of a Quadratic Imaginary Field. Documenta Mathematica 11 (2006), 73–118. | MR

[3] A. Brumer, On the units of algebraic number fields. Mathematika 14 (1967), 121–124. | MR | Zbl

[4] J. Coates and A. Wiles, On p-adic l-functions and elliptic units. J. Australian Math. Soc. 26, (1978), 1–25. | MR | Zbl

[5] E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication. Perspectives in Mathematics 3, Academic Press, 1987. | MR | Zbl

[6] R. Gillard, Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes. J. Reine Angew. Math. 358 (1985), 76–91. | MR | Zbl

[7] C. Greither, Class groups of abelian fields, and the main conjecture. Annal. Inst. Fourier 42 (1992), 445–499. | Numdam | MR | Zbl

[8] K. Iwasawa, Lectures on p-adic L-functions. Princeton University Press, 1972. | MR | Zbl

[9] S. Lang, Cyclotomic Fields I and II. Springer-Verlag, 1990. | MR | Zbl

[10] Neukirch, Schmidt and Wingberg, Cohomology of Number Fields. Springer-Verlag, 2000. | MR

[11] H. Oukhaba, On Iwasawa theory of elliptic units and 2-ideal class groups. Prépub. lab. Math. Besançon (2010).

[12] G. Robert, Unités elliptiques. Bull. soc. math. France 36, (1973). | Numdam | MR | Zbl

[13] G. Robert, Unités de Stark comme unités elliptiques. Prépub. Inst. Fourier 143, (1989).

[14] G. Robert, Concernant la relation de distribution satisfaite par la fonction ϕ associée à un réseau complexe. Inven. math. 100 (1990), 231–257. | MR | Zbl

[15] K. Rubin, The ”main conjectures” of Iwasawa theory for imaginary quadratic fields. Inven. math. 103 (1991), 25–68. | MR | Zbl

[16] K. Rubin, More ”Main Conjectures” for Imaginary Quadratic Fields. Centre Rech. Math. 4 (1994), 23–28. | MR | Zbl

[17] S. Viguié, On the classical main conjecture for imaginary quadratic fields. Prépub. lab. Math. Besançon (2011).

[18] S. Viguié, Global units modulo elliptic units and ideal class groups. to appear in Int. J. Number Theory.

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