On the Birch-Tate conjecture for cyclic number fields

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

We determine all the simplest cubic fields whose ideal class groups have exponent dividing 3, thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number 1 and the determination by D. Byeon of all all the simplest cubic fields with class number 3. We prove that there are 23 simplest cubic fields with ideal class groups of exponent 3 (and 8 simplest cubic fields with ideal class groups of exponent 1, i.e. with class number one).

In this paper, we consider a family of cubic fields $\{K_m\}_{m\geq4}$ associated to the irreducible cubic polynomials $P_m(x)=x^3-mx^2-(m+1)x-1,\,\,\,(m\geq4).$ We prove that there are infinitely many $\{K_m\}_{m\geq4}$'s whose class numbers are divisible by a given integer n. From this, we find that there are infinitely many non-normal totally real cubic fields with class number divisible by any given integer n.

The aim of this series of papers is to study cyclic extensions of the field of rational numbers of odd prime power degree. This paper, which is the third of this series, concentrates on the situation where some of the ramified primes are (partially) decomposed.

This paper explores the leading terms of equivariant Artin L -functions at $$s=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for finite abelian extensions F / k , where k is an imaginary quadratic number field. We introduce a module constructed from these leading terms and the Dirichlet regulator, and compute its generalized index relative to the torsion-free part of the unit group of F . The main result establishes that this index equals the ratio of class numbers $$h_F/h_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> multiplied by an explicit annihilator index involving roots of unity. For primes p not dividing the Galois group order, we provide a p -adic refinement using idempotents from higher Stark theory. As consequences, we derive obstructions to the existence of integral higher Stark elements mapping to the equivariant L -value and prove class number divisibility under non-decomposition conditions for ramified places. These findings contribute to the verification of equivariant Stark-type conjectures and offer new insights into class number relations in abelian extensions.

In this paper the author describes the construction of a table of totally real cubic number fields. Each field is distinguished by the coefficients of a generating polynomial, the index of this polynomial over the field and the discriminant of the field. The class number and a fundamental pair of units is also given.

Using Cohen's construction of defining polynomials for a cyclic group of odd prime order, we define a polynomial with some parameters which generates cyclic extensions of a given odd prime degree, and prove it to be generic in the sense as defined below.

Simultaneous asymptotic diophantine approximations to a basic of a real cubic number field

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 is 10921 = 67 ⋅ 163 10921 = 67 \cdot 163 , which is the conductor of the biquadratic number field Q ( − 67 , − 163 ) {\mathbf {Q}}(\sqrt { - 67} ,\sqrt { - 163} ) .

We prove a stronger version of the keystone result of Dasgupta and Kakde (Ann Math 197:289–388, 2023) on the $$\mathbb Z[G(H/F)]^-$$ Z [ G ( H / F ) ] - -Fitting ideals of certain Selmer modules $$Sel_S^T(H)^-$$ S e l S T ( H ) - associated with an abelian, CM extension H / F of a totally real number field F , and use this to compute the $$\mathbb Z_p[[G(H_\infty /F)]]^-$$ Z p [ [ G ( H ∞ / F ) ] ] - -Fitting ideal of the Iwasawa module analogues $$Sel_S^T(H_\infty )_p^-$$ S e l S T ( H ∞ ) p - of these Selmer modules, where $$H_\infty $$ H ∞ is the cyclotomic $${\mathbb {Z}}_p$$ Z p -extension of H , for an odd prime p . Our main Iwasawa theoretic result states that the $$\mathbb Z_p[[G(H_\infty /F)]]^-$$ Z p [ [ G ( H ∞ / F ) ] ] - -module $$Sel_S^T(H_\infty )_p^-$$ S e l S T ( H ∞ ) p - is of projective dimension 1 (unlike the $${\mathbb {Z}}[G(H/F)]^-$$ Z [ G ( H / F ) ] - -module $$Sel_S^T(H)^-$$ S e l S T ( H ) - which could have infinite projective dimension), is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant p -adic L -function $$\Theta _S^T(H_\infty /F)$$ Θ S T ( H ∞ / F ) . Further, we establish a perfect duality pairing between $$Sel_S^T(H_\infty )_p^-$$ S e l S T ( H ∞ ) p - and a certain $$\mathbb Z_p[[G(H_\infty /F)]]^-$$ Z p [ [ G ( H ∞ / F ) ] ] - -module $${\mathcal {M}}_S^T(H_\infty )^-$$ M S T ( H ∞ ) - , essentially introduced by Greither and the second author in (J Algebraic Geom 24:629–692, 2015). As a consequence, we recover the Equivariant Main Conjecture for the Tate module $$T_p(\mathcal M_S^T(H_\infty ))^-$$ T p ( M S T ( H ∞ ) ) - , proved in loc.cit. under the hypothesis that the classical Iwasawa $$\mu $$ μ -invariant associated with H and p vanishes. As a further consequence, we give an unconditional proof of the refined Coates–Sinnott Conjecture, proved in loc.cit. under the same $$\mu =0$$ μ = 0 hypothesis, and also proved unconditionally but with different methods by Johnston and Nickel in (An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications, arxiv:2010.03186), regarding the $$\mathbb Z[G(H/F)]$$ Z [ G ( H / F ) ] -Fitting ideals of the higher Quillen K -groups $$K_{2n-2}({\mathcal {O}}_{H,S})$$ K 2 n - 2 ( O H , S ) , for all $$n\ge 2$$ n ≥ 2 . Finally, we combine the techniques developed in the process with the method of “Taylor–Wiles primes” (introduced by Wiles (Ann Math (2) 131: 555–565, 1990) and refined by Greither in (Math Z 233: 515–534, 2000)) to strengthen further the keystone result in Dasgupta and Kakde (Ann Math 197:289–388, 2023) and prove, as a consequence, a conjecture of Burns–Kurihara–Sano on Fitting ideals of Selmer groups of CM number fields.

We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. Moreover, the 2-regular case is precisely the class of totally real number fields that have homotopy cartesian “Bökstedt square”, relating the K-theory of the 2-integers to that of the fields of real and complex numbers and finite fields. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8 in the orthogonal case. In both the orthogonal and symplectic cases, we prove a 2-primary hermitian homotopy limit conjecture for these rings.

We consider the problem of designing reliable and confidential lattice codes, known as wiretap lattice codes, for fast fading channels. We identify as code design criterion for finite lattice constellations a finite sum of inverse of algebraic norms, in fact the analogous for finite sums of the known criterion for infinite constellations, which motivates us to study totally real number fields with few elements of small norms, over which ideal lattices can be built to provide wiretap lattice codes. More precisely, we first narrow down our search to Abelian number fields known to provide reliable lattice codes, among which we look for no small inert primes and large regulators.

An algebraic description of basic discrete symmetries (space reversal P, time reversal T, and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah–Bott–Shapiro periodicity. Generalized Brauer–Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford–Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family $$ {\mathcal{K}}_n $$ , where K ∈ $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ $$ {\mathcal{K}}_n $$ with $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)].

Let p be an odd prime. Let $\mathcal{G}$ be a compact p-adic Lie group with a quotient isomorphic to ℤ p . We give an explicit description of K 1 of the Iwasawa algebra of $\mathcal{G}$ in terms of Iwasawa algebras of Abelian subquotients of $\mathcal{G}$ . We also prove a result about K 1 of a certain canonical localisation of the Iwasawa algebra of $\mathcal{G}$ , which occurs in the formulation of the main conjectures of noncommutative Iwasawa theory. These results predict new congruences between special values of Artin L-functions, which we then prove using the q-expansion principle of Deligne-Ribet. As a consequence we prove the noncommutative main conjecture for totally real fields, assuming a suitable version of Iwasawa’s conjecture about vanishing of the cyclotomic μ-invariant. In particular, we get an unconditional result for totally real pro-p p-adic Lie extension of Abelian extensions of ℚ.