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Hyperbolic tessellations and generators of
${K}_{\textbf {3}}$ for imaginary quadratic fields
- Part of
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- Journal:
- Forum of Mathematics, Sigma / Volume 9 / 2021
- Published online by Cambridge University Press:
- 24 May 2021, e40
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- Article
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We develop methods for constructing explicit generators, modulo torsion, of the
$K_3$-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic
$3$-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite
$K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for
$ K_3 $ of any field, predict the precise power of
$2$ that should occur in the Lichtenbaum conjecture at
$ -1 $ and prove that this prediction is valid for all abelian number fields.
A database of genus-2 curves over the rational numbers
- Part of
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- Journal:
- LMS Journal of Computation and Mathematics / Volume 19 / Issue A / 2016
- Published online by Cambridge University Press:
- 26 August 2016, pp. 235-254
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We describe the construction of a database of genus-
$2$ curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated
$L$-function. This data has been incorporated into the
$L$-Functions and Modular Forms Database (LMFDB).