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On certain closed subgroups of SL(2, ℤP[[X]])
Schlagworte:
Closed subgroups, Special linear group, Iwasawa algebra, 2000 Mathematics Subject Classification, Primary: 15A33, 15A54, Secondary: 11F80 (en)Downloads
Abstract. Let p > 2 be a prime number and let Λ = ℤP[[X]] be the ring of power series with p-adic integer coefficients. The special linear group of matrices SL(2, Λ) is equipped with several natural projections. In particular, let πx : SL(2, Λ) → SL(2, ℤP) be the natural projection which sends X ↦ 0. Suppose that G is a subgroup of SL(2, Λ) such that the projection H = πx (G) is known. In this note, different criteria are found which guarantee that the subgroup G of SL(2, Λ) is “as large as possible”, i.e. G is the full inverse image of H. Criteria of this sort have interesting applications in the theory of Galois representations.
Sea p > 2 un primo y Λ = ℤP[[X]] el anillo de series de potencias con coeficientes enteros p-adicos. El grupo lineal de matrices especial SL(2, Λ) es equipado con varias proyecciones naturales. En particular, πx : SL(2, Λ) → SL(2, ℤP) es la proyección natural que envia X ↦ 0. Suponga que G es un subgrupo de SL(2, Λ) tal que la proyección H = πx (G) es conocida. En este artículo se establecen diferentes criterios que garantizan que el subgrupo G de SL(2, Λ) es “tan grande como es posible” ; esto es, G es la imagen inversa total de H. Criterios de esta naturaleza tienen importantes aplicaciones a la teoría de representaciones de Galois.
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