On boundedness of periods of self maps of algebraic varieties

Abstract Let X be a proper scheme of finite type over $$\mathbb {Q}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> and let $$f : X \rightarrow X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> be a morphism. When K is a finite extension of $$\mathbb {Q}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> , the boundedness of f -periods in X ( K ) is a well studied question. In this note, we study the boundedness of f -periods in X ( K ) when K is an (possibly infinite) algebraic extension of $$\mathbb {Q}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> with finite residue field.