Hideyo Sasaki

Affiliation: Otemae College, Itami, Japan

Email: hsasaki@otemae.ac.jp

Title Of Talk: Quaternary universal forms over \bf{Q}$(\sqrt{13})$

Abstract: Let $F=${\bf{Q}}$(\sqrt{m})$ be a real quadratic field over {\bf{Q}} with $m$ a square-free positive rational integer and $\cal{O}$ be the integer ring in $F$. A totally positive definite integral $n$-ary quadratic form $f = f(x_1,\ldots , x_n)= \sum_{1\leq i , j \leq n} \alpha_{ij}x_{i}x_{j}$ ( $\alpha_{ij}=\alpha_{ji}\in \cal{O}$ ) is called universal if $f$ represents all totally positive definite integers in $\cal{O}$. Chan-Kim-Raghavan proved that ternary universal forms over $F$ exist if and only if $m=2 , 3 , 5$ and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12. We give the result that there are only two quaternary universal forms (up to equivalence) over {\bf Q}$(\sqrt{13})$. For the proof of universality we apply the theory of quadratic lattices.

Last update made Tue Feb 17 09:09:22 EST 2009.
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