arXiv subjects

Henri Cohen

Publications and source records attributed to Henri Cohen.

27 records · Page 2Linked to original sources

arXiv2017

On $D_\ell$-extensions of odd prime degree $\ell$

Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_{\ell}$-extensions of odd prime degree $\ell$ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the Ankeny-Artin-Chowla conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other topics. We also obtain improved upper bounds for the number of such extensions (over Q) of bounded discriminant.

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arXiv2016

Finite hypergeometric functions

Finite hypergeometric functions are complex valued functions on finite fields which are the analogue of the classical analytic hypergeometric functions. From the work of N.M.Katz it follows that their values are traces of Frobenius on certain l-adic sheafs. More concretely, in many instances their values can be used to give formulas for pointcounts of F_q-rational points on certain varieties. In this paper we work out the case of one-variable functions whose monodromy in the analytic case can be defined over the rational integers.

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arXiv2015

Elementary 3-descent with a 3-isogeny

In this expository paper, we show how to use in practice 3-descent with a 3-isogeny to find an estimate for the rank of an elliptic curve having a rational 3-torsion subgroup, and we also give a geometric interpretation of these computations.

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arXiv2015

Identities for field extensions generalizing the Ohno-Nakagawa relations

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of `extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper we generalize their result by proving a similar identity relating certain degree l fields with Galois groups D_l and F_l respectively, for any odd prime l, and in particular we give another proof of the Ohno-Nakagawa relation without appealing to binary cubic forms.

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arXiv2013

Dirichlet series associated to cubic fields with given quadratic resolvent

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. In the present paper we carry the analysis further and prove explicit formulas for these Dirichlet series over Q. In a companion paper we do the same for quartic fields having a given cubic resolvent. As an application (not present in the initial version), we compute tables of the number of S_3-sextic fields E with |Disc(E)| < X, for X ranging up to 10^23. An accompanying PARI/GP implementation is available from the second author's website.

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arXiv2013

Dirichlet series associated to quartic fields with given resolvent

Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series $\sum_K|\Disc(K)|^{-s}$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work. This is a companion to a previous paper where we compute the Dirichlet series associated to cubic fields having a given quadratic resolvent.

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arXiv2010

Counting Cubic Extensions with given Quadratic Resolvent

Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^α)$, for an explicit $α<1$.

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arXiv2008

Computational class field theory

Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.

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arXiv2003

Constructing and counting number fields

In this paper we give a survey of recent methods for the asymptotic and exact enumeration of number fields with given Galois group of the Galois closure. In particular, the case of fields of degree up to 4 is now almost completely solved, both in theory and in practice. The same methods also allow construction of the corresponding complete tables of number fields with discriminant up to a given bound.

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