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Explicit bounds for generators of the class group

Authors: Loïc Grenié and Giuseppe Molteni
Journal: Math. Comp. 87 (2018), 2483-2511
MSC (2010): Primary 11R04; Secondary 11R29
Published electronically: November 16, 2017
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Abstract: Assuming Generalized Riemann's Hypothesis, Bach proved that the class group $ \mathcal C\!\ell _{\mathbf {K}}$ of a number field $ \mathbf {K}$ may be generated using prime ideals whose norm is bounded by $ 12\log ^2\Delta _{\mathbf {K}}$, and by $ (4+o(1))\log ^2\Delta _{\mathbf {K}}$ asymptotically, where $ \Delta _{\mathbf {K}}$ is the absolute value of the discriminant of $ \mathbf {K}$. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates $ \mathcal C\!\ell _{\mathbf {K}}$ and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that $ \mathcal C\!\ell _{\mathbf {K}}$ is generated by prime ideals whose norm is bounded by the minimum of $ 4.01\log ^2\Delta _{\mathbf {K}}$, $ 4\big (1+\big (2\pi e^{\gamma })^{-N_{\mathbf {K}}}\big )^2\log ^2\Delta _{\mathbf {K}}$ and $ 4\big (\log \Delta _{\mathbf {K}} +\log \log \Delta _{\mathbf {K}}-(\gamma +\l... ...\frac {\log (7\log \Delta _{\mathbf {K}})} {\log \Delta _{\mathbf {K}}}\big )^2$. Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size $ \asymp (\log \Delta _{\mathbf {K}}\log \log \Delta _{\mathbf {K}})^2$. In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than $ \log ^2\Delta _{\mathbf {K}}$ except for $ 7$ out of the $ 31292$ fields we tested.

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Loïc Grenié
Affiliation: Dipartimento di Ingegneria gestionale, dell’informazione e della produzione, Università degli Studi di Bergamo, viale Marconi 5, 24044 Dalmine (BG) Italy

Giuseppe Molteni
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italy

Received by editor(s): July 8, 2016
Received by editor(s) in revised form: October 14, 2016, January 30, 2017, and March 29, 2017
Published electronically: November 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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