On -invariants attached to cyclic cubic number fields

Let F be a cubic cyclic field with t (⩾ 2) ramified primes. For a finite abelian group G, let r3(G) be the 3-rank of G. If 3 does not ramify in F, then it is proved that t−1 ⩽ r3(K2OF) ⩽ 2t. Furthermore, if t is fixed, for any s satisfying t−1 ⩽ s ⩽ 2t−1, there is always a cubic cyclic field F with exactly t ramified primes such that r3(K2OF) = s. It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula $$d_{\infty ,r} = 3^{ - r^2 } {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } \mathord{\left/ {\vphantom {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}} \right. \kern-\nulldelimiterspace} {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}^2 .$$ This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.

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).

Let [Formula: see text] be a number field and [Formula: see text] a rational prime. We obtain a simple descent criterion characterizing those projective Galois representations [Formula: see text] for which the corresponding twist [Formula: see text] of the principal modular curve of level [Formula: see text] is defined over [Formula: see text]. We also give a more concrete version of this result for representations which arise from elliptic curves over cyclic number fields.

Galois 2-extensions unramified outside 2

The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve – which we refer to as "inter-universal Teichmüller theory" – by applying the theory of semi-graphs of anabelioids , Frobenioids , the étale theta function , and log-shells developed in earlier papers by the author. We begin by fixing what we call "initial \Theta -data" , which consists of an elliptic curve E_F over a number field F , and a prime number l\ge 5 , as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite étale coverings to the once-punctured elliptic curve X_F determined by E_F . These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring \mathbb{F}_l=\mathbb{Z}/l\mathbb{Z} acting on the l -torsion points of the elliptic curve. We then construct " \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters" associated to the given \Theta -data. These \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field – which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field – are, in some sense, "dismantled" or "disentangled" from one another. All \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters are isomorphic to one another, but may also be related to one another by means of a " \Theta -link" , which relates certain Frobenioid-theoretic portions of one \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater to another in a fashion that is not compatible with the respective conventional ring/scheme theory structures . In particular, it is a highly nontrivial problem to relate the ring structures on either side of the \Theta -link to one another. This will be achieved, up to certain "relatively mild indeterminacies" , in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an "alien ring structure" [associated, say, to the domain of the \Theta -link] in terms of a given ring structure [associated, say, to the codomain of the \Theta -link] will be applied in the final paper of the series to obtain results in diophantine geometry . Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a p -adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.

On Steinitz classes of nonabelian Galois extensions and p-ary cyclic Hamming codes

Let p be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a \mathbb{Z}_{p} -tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain \mathbb{Z}_{p} -extensions in which a given prime above p is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for p -primary class numbers of the number fields in a \mathbb{Z}_{p} -tower.

Structure galoisienne relative des anneaux d'entiers

Let p be an odd prime number and K a number field having a primitive p th root of unity \zeta_p . We prove that Nikshych's non group-theoretical Hopf algebra H_p , which is defined over \mathbb Q(\zeta_p) , admits a Hopf order over the ring of integers \mathcal O_K if and only if there is an ideal I of \mathcal O_K such that I^{2(p-1)} = (p) . This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over \mathcal O_K exists, it is unique and we describe it explicitly.

Let K be an imaginary cyclic quartic number field whose 2-class group is nontrivial, it is known that there exists at least one unramified quadratic extension F of K. In this paper, we compute the rank of the 2-class group of the field F.

Irreducible strongly indecomposable torsion-free abelian groups of finite rank are called Reid groups, and a full subring R of a division algebra D is called Reid-realizable if it is isomorphic to the endomorphism ring of a Reid group. The main results is that a full subring R of a finite dimensional division algebra D is Reid-realizable if for infinitely many prime numbers p the quotient of the p-adic localizations D p * / div R p * > either is the product of at least two p-adic number fields, or is not a product of division algebras. In particular, a full subring of a finite dimensional central division algebra over an algebraic number field is Reid-realizable if it is locally free for infinitely many prime numbers.

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.

Integer factoring is a curious number theory problem with wide applications in complexity and cryptography. The best known algorithm to factor a number n takes time, roughly, exp(2*log^{1/3}(n)*log^{2/3}(log(n))) (number field sieve, 1989). One basic idea used is to find two squares, possibly in a number field, that are congruent modulo n. Several variants of this idea have been utilized to get other factoring algorithms in the last century. In this work we intend to explore new ideas towards integer factoring. In particular, we adapt the AKS primality test (2004) ideas for integer factoring. In the motivating case of semiprimes n=pq, i.e. p<q are primes, we exploit the difference in the two Frobenius morphisms (one over F_p and the other over F_q) to factor n in special cases. Specifically, our algorithm is polynomial time (on number theoretic conjectures) if we know a small algebraic dependence between p,q. We discuss families of n where our algorithm is significantly faster than the algorithms based on known techniques.

Ternary quadratic forms over number fields with small class number