Resumen de On certain closed subgroups SL(2,Z[[X]])

Álvaro Lozano Robledo

• español

  Sea [p] > 2 un primo y [LAMBDA] = [Z_p][X] el anillo de series de potencias con coeficientes enteros [p]-adicos. El grupo lineal de matrices especial SL(2, LAMBDA) es equipado con varias proyecciones naturales. En particular, [pi_X]: SL(2, LAMBDA) ---> SL(2, [Z_p]) es la proyección natural que envia [X] ---> 0. Suponga que [G] es un subgrupo de SL(2, LAMBDA) tal que la proyección [H] = [pi_X (G)] es conocida. En este artículo se establecen diferentes criterios que garantizan que el subgrupo [G] de SL(2, LAMBDA) 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.

• English

  Let [p] > 2 be a prime number and let [LAMBDA] = [Z_p][X] be the ring of power series with [p]-adic integer coefficients. The special linear group of matrices SL(2, LAMBDA) is equipped with several natural projections. In particular, [pi_X]: SL(2, LAMBDA) ---> SL(2, [Z_p]) be the natural projection which sends [X] ---> 0. Suppose that [G] is a subgroup of SL(2, LAMBDA) such that the projection [H] = [pi_X (G)] is known. In this note, different criteria are found which guarantee that the subgroup [G] de SL(2, LAMBDA) 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.