## An $L(1/3)$ algorithm for ideal class group and regulator computation in certain number fields

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## Abstract:

We analyze the complexity of the computation of the class group structure, regulator, and a system of fundamental units of an order in a certain class of number fields. Our approach differs from Buchmann’s, who proved a complexity bound under the generalized Riemann hypothesis of $L(1/2,O(1))$ when the discriminant tends to infinity with fixed degree. We achieve a heuristic subexponential complexity in $O(L(1/3,O(1)))$ under the generalized Riemann hypothesis when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge, Gaudry and Thomé in the context of algebraic curves over finite fields. We also address rigorously the problem of the precision of the computation of the regulator.## References

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## Additional Information

**Jean-François Biasse**- Affiliation: LIX, École Polytechnique, 91128 Palaiseau, France
- Email: biasse@lix.polytechnique.fr
- Received by editor(s): December 17, 2009
- Received by editor(s) in revised form: October 25, 2011
- Published electronically: January 17, 2014
- Additional Notes: The author was supported by a DGA grant
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp.
**83**(2014), 2005-2031 - MSC (2010): Primary 54C40, 14E20; Secondary 46E25, 20C20
- DOI: https://doi.org/10.1090/S0025-5718-2014-02651-3
- MathSciNet review: 3194139