The Multinorm Principle

A finite extension of global fields is said to satisfy the Hasse norm principle if K × ∩N L/K (J L ) = N L/K (L × ), where N L/K : J L → J K 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 L 1 and L 2 of a global field K satisfies the multinorm principle if K × ∩N L1 /K(J L1 )N L2 /K(J L2 ) = N L1 /K(L × 1)N L2 /K(L × 2). The obstruction to the multinorm principle is defined to be the quotient group X(L 1 ,L 2 /K) := K × ∩N L1 /K(J L1 )N L2 /K(J L2 )/N L1 /K(L × 1)N L2 /K(L × 2). In this work, we analyze the multinorm principle and compute X(L 1 ,L 2 /K) in several important special cases. In particular, we show that the multinorm principle always holds when L 1 and L 2 are separable extensions of K with linearly disjoint Galois closures, and we prove that X(L 1 ,L 2 /K) = X(L 1 ∩L 2 /K) when L 1 and L 2 are abelian extensions of K. We give a partial description of the obstruction X(L 1 ,L 2 /K) in terms of group cohomology and class field theory by relating X(L 1 ,L 2 /K) to the Tate-Shafarevich groups X(L i /K) for i = 1,2. Then, in the special cases mentioned above, we show how this description can be exploited to compute X(L 1 ,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. ii Finally, we identify X(L 1 ,L 2 /K) with X(T), where T is the multinorm torus associated to L 1 and L 2 , and use a cohomological argument to prove that X(T) vanishes for certain families of extensions. In particular, X(L 1 ,L 2 /K) vanishes whenever L 1 and L 2 are Galois extensions of K and L 1 ∩L 2 is a cyclic extension of K.

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