Eisenstein Cohomology for $\mathrm {GL}_N$ and the special values of Rankin–Selberg L-functions over a totally imaginary number field

It is a classical result, deriving from the Gaussian theory of genera of integral binary quadratic forms, that there exist only finitely many imaginary quadratic fields for which the ideal class group is a group of exponent two. This finiteness has been shown to extend to all those totally imaginary quadratic extensions of any fixed totally real algebraic number field. In this paper we put forward the conjecture that there exist only finitely many imaginary abelian algebraic number fields which have ideal class groups of exponent two, and we examine the extent to which existing methods can be brought to bear on this conjecture. One consequence of the validity of the conjecture would be a proof of the existence of finite abelian groups which do not occur as the ideal class group of any imaginary abelian field.

Assuming that the list of imaginary quadratic number fields of class-number 4 is complete, a determination is made of all imaginary bicyclic biquadratic number fields of class-number 2.

Let \(k=Q({\sqrt d,\sqrt{-q}})\) be an imaginary biquadratic number field with Clk,2, the 2-class group of k, isomorphic to Z/2Z × Z/2mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1.

In 2024, Ram proved that the class number of an imaginary cyclic quartic number field is never equal to a prime [Formula: see text]. Here, we greatly generalize this result to the case of the non-quadratic imaginary cyclic number fields of [Formula: see text]-power degrees and not necessarily prime class numbers.

We determine all nonquadratic imaginary cyclic number fieldsKof 2-power degrees with ideal class groups of exponents≤2\leq 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some(Z/2Z)m,m≥0{({\mathbf {Z}}/2{\mathbf {Z}})^m},m \geq 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ones (see Theorem 12), and 1 of them has degree 16 (see Theorem 11).

CM-Fields with Cyclic Ideal Class Groups of 2-Power Orders

Leopoldt's conjecture for imaginary Galois number fields

We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.

Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents ≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301 such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination.

Class Number Problem for Imaginary Cyclic Number Fields

Uchida proved that there exist only finitely many imaginary belian number fields with class number one [12], and gave the value 2 10 as an upper bound of the conductors of such fields [13].Several authors deter- mined such fields of some types, but not all of them have been determined yet.(Masley determined the cyclotomic number fields with class number one [8], and Uchida determined such fields of two power degrees [14].)"The purpose of this article is to report that we have determined all the imaginary abelian number fields with class number one in proving the fol- lowing.(The details will appear elsewhere [15].)Theorem.There exist exactly 171 imaginary abelian number fields with class number one as given in the attached table.Among them, 29 fields are cyclotomic, 49 fields are cyclic, and 87 fields are maximal with respect to inclusion.The maximal conductor of these fields is 10921= 67.163, which is the conductor of the biquadratic number field Q(/-67, /--163).Now we sketch here the method of proof.The basic idea is due to Uchida.(See [13] [14].)In the following, let K be an imaginary abelian number fields with class number h(K).When h(K)--l, the genus numbeV of K is one, which is equivalent to say that the character group X corres- ponding to K is a direct product of subgroups generated by a character of prime power conductor" X--(Z)Zr), the conductor f of --a prime power.Moreover, when h(K)--1, the subfield of K corresponding to (Z} has strict class number one ([13], Prop.1).Therefore, we need to consider only K of this type.Among K of this type, we first determine K with h-(K)----1, and then check whether h/(K)=l, or not.Here, h-(K) (resp.h/(K)) de- notes the relative class number of K (resp.the class number of the maxi- mal real subfield of K). (h(K)--h-(K)h+(K).)The key idea which facili- tates the determination is" ( ) If h(K)-l, then, for any subfield F of K such that K/F is totally ramified at a finite prime, the strict class number of F is one.From this, in most cases, we can immediately restrict K to be considered 1)Recently, Louboutin determined imaginary abelian sextic number fields with class number one [4].The author knew his result after completion o.

On the second 2-class group [formula omitted] of some imaginary quartic cyclic number field K

Let N be an imaginary abelian number field. We know that h-, the relative class number of N, goes to infinity as fN, the conductor of N, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CMfields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Second, we have proved in this paper that there are exactly 48 such fields.

We develop an efficient technique for computing relative class numbers of imaginary abelian fields, efficient enough to enable us to easily compute relative class numbers of imaginary cyclic fields of degrees 32 and conductors greater than 1013, or of degrees 4 and conductors greater than 1015. Acccording to our extensive computation, all the 166204 imaginary cyclic quartic fields of prime conductors p less than 107 have relative class numbers less than p/2. Our major innovation is a technique for computing numerically root numbers appearing in some functional equations.

In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is10921=67⋅16310921 = 67 \cdot 163, which is the conductor of the biquadratic number fieldQ(−67,−163){\mathbf {Q}}(\sqrt { - 67} ,\sqrt { - 163} ).