The article proposes an algorithm for decoding and representation in natural language of the Universal Decimal Classifycation (UDC) complex class numbers. The algorithm is based on the formal ...definition of correct class numbers using a generative grammar, which sets the list of structures starting with simple table codes of UDC classes. Then separate integers, auxiliary and independent class numbers are sequentially attached to the codes with special symbols of relations of classes, which compose the complex class number. The algorithm expresses the values of the analyzed complex indices by descriptions (names and notes) of the table classes included in the structure of the analyzed string. The class descriptions are accompanied with the logical connectors based on the functions of the auxiliary characters. They provide the idea on connection of concepts denoted in the class number. The algorithm action is described evidently for the analysis of combined index 539.4.019: 535-15+537.8.029.6. The proposed algorithm is applicable both to visualize the meaning of complex class numbers, and to ensure the completeness and accuracy of the documents retrieval by the UDC classes.
In this paper, we study moments of Hurwitz class numbers associated to imaginary quadratic orders restricted into fixed arithmetic progressions. In particular, we fix
t
t
in an arithmetic progression
...t
≡
m
(
mod
M
)
t\equiv m\ \, \left ( \operatorname {mod} \, M \right )
and consider the ratio of the
2
k
2k
-th moment to the zeroeth moment for
H
(
4
n
−
t
2
)
H(4n-t^2)
as one varies
n
n
. The special case
n
=
p
r
n=p^r
yields as a consequence asymptotic formulas for moments of the trace
t
≡
m
(
mod
M
)
t\equiv m\ \, \left ( \operatorname {mod} \, M \right )
of Frobenius on elliptic curves over finite fields with
p
r
p^r
elements.
In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of ...a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes t_j\mathbb {Z}^2) relating the number of lattice points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.
For positive rank r elliptic curves E(Q), we employ ideal class pairingsE(Q)×E−D(Q)→CL(−D), for quadratic twists E−D(Q) with a suitable “small y-height” rational point, to obtain explicit class ...number lower bounds that improve on earlier work by the authors. For the curves E(a):y2=x3−a, with rank r(a), this givesh(−D)≥110⋅|Etor(Q)|RQ(E)⋅πr(a)22r(a)Γ(r(a)2+1)⋅log(D)r(a)2loglogD, representing a general improvement to the classical lower bound of Goldfeld, Gross and Zagier when r(a)≥3. We prove that the number of twists E−D(a)(Q) with such a suitable point (resp. with such a point and rank ≥2 under the Parity Conjecture) is ≫a,εX12−ε. We give infinitely many cases where r(a)≥6. These results can be viewed as an analogue of the classical estimate of Gouvêa and Mazur for the number of rank ≥2 quadratic twists, where in addition we obtain “log-power” improvements to the Goldfeld-Gross-Zagier class number lower bound.
Ideal class pairings map the rational points of rank r≥1 elliptic curves E/Q to the ideal class groups CL(−D) of certain imaginary quadratic fields. These pairings imply thath(−D)≥12(c(E)−ε)(logD)r2 ...for sufficiently large discriminants −D in certain families, where c(E) is a natural constant. These bounds are effective, and they offer improvements to known lower bounds for many discriminants.
We present a formula for the class number of a multinorm one torus TL/k associated to any étale algebra L over a global field k. This is deduced from a formula for analogues of invariants introduced ...by T. Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono's invariant for arbitrary S-ideal class numbers and the narrow version, generalizing results of Katayama, Morishita, Ono and Sasaki.
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