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arXiv · 1904.03466

A p-adic analogue of Siegel's Theorem on sums of squares

Sylvy Anscombe·Philip Dittmann·Arno Fehm
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Abstract

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded.

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Sylvy Anscombe, Philip Dittmann, Arno Fehm. 2019-04-06. A p-adic analogue of Siegel's Theorem on sums of squares. https://doi.org/10.1002/mana.201900173

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