A generalized Mazur’s theorem and its applications

Abstract

We generalize a theorem of Mazur concerning the universal norms of an abelian variety over a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript p Superscript d"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>d</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathbb {Z}_p^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -extension of a complete local field. Then we apply it to the proof of a control theorem for abelian varieties over global function fields.