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arXiv · 2212.09461

Breaking the 4 barrier for the bound of a generating set of the class group

Loïc Grenié·Giuseppe Molteni
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Abstract

Let $K$ be a field of degree $n$ and discriminant with absolute value $\Delta$. Under the assumption of the validity of the Generalized Riemann Hypothesis, we provide a new algorithm to compute a set of generators of the class group of $K$ and prove that the norm of the ideals in that set is $\leq (4-1/(2n))\log^2\Delta$, except for a finite number of fields of degree $n\leq 4$. For those fields, the conclusion holds with the slightly larger limit $(4-1/(2n)+1/(2n^2))\log^2\Delta$. When the cardinality of $\mathcal C\!\ell$ is odd the bounds improve to $(4-2/(3n))\log^2\Delta$, again with finitely many exceptions in degree $n\leq 4$, and to $(4-2/(3n)+3/(8n^2))\log^2\Delta$ without exceptions.

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Loïc Grenié, Giuseppe Molteni. 2022-12-19. Breaking the 4 barrier for the bound of a generating set of the class group. https://arxiv.org/abs/2212.09461

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