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indefinite quadratic form in four variables universal over p-adic integers

数论 Math StackExchange 1 票 1 回答 43 浏览 提问者: OEG 2026-06-25 12:18
number-theory reference-request quadratic-forms p-adic-number-theory

问题内容

I need a source for the following statement:

Let $q(x,y)=ax^2+bxy+cy^2$ be a binary quadratic form with $a,b,c\in\mathbb Z$ and let $p$ be a prime with $p\not\mid 2D$, where $D=b^2-4ac$ is not a square. Then the quaternary quadratic form $q(x_1,y_1)-q(x_2,y_2)$ represents all $p$-adic integers, as $x_1,x_2,y_1,y_2$ vary over $\mathbb Z_p$.

I looked for a statement like this in [Cassels, Rational Quadratic Forms] but could'nt find anything. Thank you in advance!

回答 (1)

Dietrich Burde 0 票 2026-06-25 13:49 原文

There should be a reference in the article "On the gaps between values of binary quadratic forms" by Jörg Brüdern and Rainer Dietmann from $2011$, in the context of discussing the equation $q(y,x)-q(z,w)=k$ for integers $k$.

"Since it is non-degenerate and universally represents units, it represents all $p$-adic integers."