On the resolution of index form equations in dihedral quartic number fields.
On the verification of Clark's example of a euclidean but not norm-euclidean number field.
Periodic Gaussian moats.
Power integral bases in cubic relative extensions.
Prime ideal factorization in a number field via Newton polygons
Let be a number field defined by an irreducible polynomial and its ring of integers. For every prime integer , we give sufficient and necessary conditions on that guarantee the existence of exactly prime ideals of lying above , where factors into powers of monic irreducible polynomials in . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly prime ideals of lying above . We further specify...
Quadratic extensions of quintic fields of signature
Ray class fields of global function fields with many rational places
Regulators of rank one quadratic twists
We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.
Relaxed algorithms for -adic numbers
Current implementations of -adic numbers usually rely on so called zealous algorithms, which compute with truncated -adic expansions at a precision that can be specified by the user. In combination with Newton-Hensel type lifting techniques, zealous algorithms can be made very efficient from an asymptotic point of view.In the similar context of formal power series, another so called lazy technique is also frequently implemented. In this context, a power series is essentially a stream of coefficients,...
Sieving in function fields.
Some methods for evaluating the regulator of a real quadratic function field.
Some results from the tables of irregularity index of a prime
Standard relations of multiple polylogarithm values at roots of unity.
Stark's conjectures and Hilbert's twelfth problem.
Sur la racine carrée de la codifférente
On présente deux résultats nouveaux concernant la racine carrée de la codifférente d’une extension faiblement ramifiée de . Le premier, relatif à sa structure galoisienne, s’inscrit dans la stratégie classique développée notamment par Fröhlich et Taylor. Le second, qui concerne le réseau entier unimodulaire associé, est prouvé à l’aide de calculs numériques portant sur des exemples intéressants.
The 4-class ranks of quadratic fields related to the Weyl group algebra.
The class number one problem for some non-abelian normal CM-fields of degree
We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to , the alternating group of degree and order . There are two such fields with Galois group (see Theorem 14) and at most one with Galois group SL (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number .
The class number one problem for the dihedral and dicyclic CM-fields
We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.
The class number one problem for the non-abelian normal CM-fields of degree 24 and 40
The class number one problem for the non-normal sextic CM-fields. Part 2