2. On the Rank of Universal Quadratic Forms over Real Quadratic Fields

  • Valentin Blomer University of G\"ottingen, Mathematisches Institut, Bunsenstr.~3-5, D-37073 G\"ottingen, Germany
  • V\' \i t\v ezslav Kala University of G\"ottingen, Mathematisches Institut, Bunsenstr.~3-5, D-37073 G\"ottingen, Germany and Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolov\-sk\' a 83, 18600 Praha~8, Czech Republic
Keywords: universal quadratic form, real quadratic number field, continued fraction, additively indecomposable integer


We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction.
We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.


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