scholarly journals $GL_{2}(O_K)$-invariant lattices in the space of binary cubic forms with coefficients in the number field $K$

2014 ◽  
Vol 142 (7) ◽  
pp. 2313-2325
Author(s):  
Charles A. Osborne
Keyword(s):  
Number Field ◽  
Cubic Forms

scholarly journals A construction of quintic rings

2004 ◽  
Vol 173 ◽  
pp. 163-203 ◽  
Author(s):  
Anthony C. Kable ◽  
Akihiko Yukie
Keyword(s):  
Number Field ◽  
Upper Bound ◽  
Number Fields ◽  
Equivalence Classes ◽  
Ring Of Integers ◽  
Isomorphism Classes ◽  
Cubic Forms

AbstractWe construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

Cubic forms over algebraic number fields

1969 ◽  
Vol 66 (2) ◽  
pp. 323-333 ◽  
Author(s):  
C. Ryavec
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Rational Field ◽  
Cubic Form ◽  
Proper Subvariety ◽  
Almost All ◽  
Cubic Forms

In 1935 Tartakowski (7) proved that, in general, a cubic form in sufficiently many variables with coefficients in an algebraic number field K has a non-trivial zero in that field; and in the case when K is the rational field 57 variables suffice. Here, ‘in general’ means that the coefficients of the form do not lie in a proper subvariety of the coefficient space. Hence, Tartakowski's result holds for almost all cubic forms. Later, Lewis (5) proved that if K is any algebraic number field such that [K: Q] = n, then there exists a function ψ(n) such that every cubic form over K in m ≥ ψ(n) variables has a non-trivial zero in K. His bound, ψ(n), is extremely large; e.g. when K is the rational field, ψ(1) > 500.

Cubic forms over algebraic number fields

1963 ◽  
Vol 59 (4) ◽  
pp. 683-705 ◽  
Author(s):  
C. P. Ramanujam
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Precise Form ◽  
Cubic Form ◽  
Cubic Forms

Davenport has proved (3) that any cubic form in 32 or more variables with rational coefficients has a non-trivial rational zero. He has also announced that he has subsequently been able to reduce the number of variables to 29. Following the method of (3), we shall prove that any cubic form over any algebraic number field has a non-trivial zero in that field, provided that the number of variables is at least 54. The following is the precise form of our result.

K-STRATUM LIMIT AND K-STRATUM INTEGRAL ON THE GENERALIZED NUMBER FIELD

1982 ◽  
Vol 2 (4) ◽  
pp. 375-388
Author(s):  
Jiwu Wang ◽  
Tai Kang
Keyword(s):  
Number Field

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

scholarly journals Hydride-based antiperovskites with soft anionic sublattices as fast alkali ionic conductors

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Shenghan Gao ◽  
Thibault Broux ◽  
Susumu Fujii ◽  
Cédric Tassel ◽  
Kentaro Yamamoto ◽  
...  
Keyword(s):  
Experimental Studies ◽  
Ionic Conductivities ◽  
Cubic Phase ◽  
Ionic Conductors ◽  
Large Size ◽  
Migration Barriers ◽  
Soft Phonon Mode ◽  
Aliovalent Substitution ◽  
Cubic Forms ◽  
The Ideal

AbstractMost solid-state materials are composed of p-block anions, only in recent years the introduction of hydride anions (1s2) in oxides (e.g., SrVO2H, BaTi(O,H)3) has allowed the discovery of various interesting properties. Here we exploit the large polarizability of hydride anions (H–) together with chalcogenide (Ch2–) anions to construct a family of antiperovskites with soft anionic sublattices. The M3HCh antiperovskites (M = Li, Na) adopt the ideal cubic structure except orthorhombic Na3HS, despite the large variation in sizes of M and Ch. This unconventional robustness of cubic phase mainly originates from the large size-flexibility of the H– anion. Theoretical and experimental studies reveal low migration barriers for Li+/Na+ transport and high ionic conductivity, possibly promoted by a soft phonon mode associated with the rotational motion of HM6 octahedra in their cubic forms. Aliovalent substitution to create vacancies has further enhanced ionic conductivities of this series of antiperovskites, resulting in Na2.9H(Se0.9I0.1) achieving a high conductivity of ~1 × 10–4 S/cm (100 °C).

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