# The Multinorm Principle

Pollio, Timothy P., Department of Mathematics, University of Virginia

Rapinchuk, Andrei, Department of Mathematics, University of Virginia

Abramenko, Peter, Department of Mathematics, University of Virginia

Ershov, Mikhail, Department of Mathematics, University of Virginia

Shelat, Abhi, Department of Computer Science, University of Virginia

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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PHD (Doctor of Philosophy)

English

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2013/05/01