The Multinorm Principle.

摘要

A finite extension of global fields is said to satisfy the Hasse norm principle if Kx ∩ N L/K(JL) = NL/K( Lx), where NL/K : JL → JK denotes the natural extension of the norm map associated with L/K to the corresponding groups of ideles. By analogy, we say that a pair of finite extensions L1 and L2 of a global field K satisfies the multinorm principle if Kx ∩ NL1/K&parl0;JL1&parr0; NL2/K&parl0;JL2&parr0; =NL1/K&parl0;Lx1 &parr0;NL2/K&parl0;Lx 2&parr0; . The obstruction to the multinorm principle is defined to be the quotient group III(L1, L2 /K) := Kx ∩ NL1/K&parl0;JL1&parr0; NL2/K&parl0;JL2&parr0; /NL1/K&parl0;Lx1 &parr0;NL2/K&parl0;Lx 2&parr0; .;In this work, we analyze the multinorm principle and compute III( L1, L2/K) in several important special cases. In particular, we show that the multinorm principle always holds when L1 and L 2 are separable extensions of K with linearly disjoint Galois closures, and we prove that III(L1, L2/K) = III(L1 ∩ L2/K) when L1 and L2 are abelian extensions of K..;We give a partial description of the obstruction III(L 1, L2/K) in terms of group cohomology and class field theory by relating III(L 1, L2/K) to the Tate-Shafarevich groups III(Li/K) for i = 1, 2. Then, in the special cases mentioned above, we show how this description can be exploited to compute III(L1, L 2/K) exactly.;Additionally, we define a generalization of the multinorm principle for n-tuples of extensions (n ≥ 3). By generalizing the methods described above, we are able to prove that the multinorm principle holds for any n-tuple of finite Galois extensions of K which are linearly disjoint over K as a family.;Finally, we identify III(L1, L 2/K) with III(T), where T is the multinorm torus associated to L1 and L2, and use a cohomological argument to prove that III(T) vanishes for certain families of extensions. In particular, III(L1, L 2/K) vanishes whenever L1 and L2 are Galois extensions of K and L1 ∩ L2 is a cyclic extension of K..