Explicit Lower bounds for residues at of Dedekind zeta functions and relative class numbers of CM-fields

Let be a given set of positive rational primes. Assume that the value of the Dedekind zeta function of a number field is less than or equal to zero at some real point in the range . We give explicit lower bounds on the residue at of this Dedekind zeta function which depend on , the absolute value of the discriminant of and the behavior in of the rational primes . Now, let be a real abelian number field and let be any real zero of the zeta function of . We give an upper bound on the residue at of which depends on , and the behavior in of the rational primes . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields which depend on the behavior in of the rational primes . We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.

     

Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

On some algebraic properties of CM-types of CM-fields and their reflexes

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The purpose of this paper is to show that the reflex fields of a given CM-field K are equipped with a certain combinatorial structure that has not been exploited yet.The first theorem is on the abelian extension generated by the moduli and the b-torsion points of abelian varieties of CM-type, for any natural number b. It is a generalization of the result by Wei on the abelian extension obtained by the moduli and all the torsion points. The second theorem gives a character identity of the Artin L-function of a CM-field K and the reflex fields of K. The character identity pointed out by Shimura (1977) in [10] follows from this.The third theorem states that some Pfister form is isomorphic to the orthogonal sum of defined on the reflex fields _⊕Φ∈ΛK^∗(Φ). This result suggests that the theory of complex multiplication on abelian varieties has a relationship with the multiplicative forms in higher dimension.

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For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=IIwksVYV5YE.     

Powerful Necessary Conditions for Class Number Problems

We give a necessary condition for the ideal class group of a CM-field to be of exponent at most two. This condition enables us to drastically reduce the amount of relative class number computation for determination of the CM - fields of some types (e. g. the imaginary cyclic non -quadratic number fields of 2 - power degrees) whose ideal class groups are of exponents at most two. We also give a necessary condition for some quartic non - CM - fields to have class number one.     

On the Fontaine–Mazur Conjecture for CM-Fields

Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.     

The class number one problem for some non-abelian normal CM-fields

Let be a non-abelian normal CM-field of degree any odd prime. Note that the Galois group of is either the dicyclic group of order or the dihedral group of order We prove that the (relative) class number of a dicyclic CM-field of degree is always greater then one. Then, we determine all the dihedral CM-fields of degree with class number one: there are exactly nine such CM-fields.