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Derong Qiu

Publications and source records attributed to Derong Qiu.

18 recordsLinked to original sources
arXiv2026

Iwasawa invariants and class number parity of multi-quadratic number fields

In this paper, based mainly on the method of Iwasawa and Kida, by studying in detail the Hasse's unit index and the ramifications of prime ideals, we obtain explicit results of Iwasawa invariants $ \lambda_{2} $ of the cyclotomic $ \Z_{2}$-extensions of number fields. In particular, under Greenberg's conjecture, we obtain an explicit formula of $ \lambda_{2} $ for imaginary multi-quadratic number fields. As an application, we give a criteria of determining class number parity of multi-quadratic number fields.

math.NT↗
arXiv2023

On $ p-$Rationality of Cubic and Quartic Number Fields

In this paper, a new criterion is given to determine the $p-$rationality of some complex cubic number fields in terms of $ p-$divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are constructed,the relations between generalized $ abc-$conjecture and the $p-$rationality are discussed, from which some explicit fields satisfying Greenberg's Generalized Conjecture (GGC, for short) are obtained.

math.NT↗
arXiv2015

On algebraic congruence varieties over semirings

In this paper, we develop some foundations for a theory of algebraic varieties of congruences on commutative semirings. By studying the structure of congruences, firstly, we show that the spectrum $ \text{Spec}^{c}(A) $ consisting of prime congruences on a semirings $ A $ has a Zariski topological structure; Then, for two semirings $ A \subset B, $ we consider the polynomial semiring $ S = A[x_{1}, \cdots , x_{n}] $ and the affine $ n-$space $ B^{n}. $ For any congruence $ \sigma $ on $ S $ and congruence $ \rho $ on $ B, $ we introduce the $ \rho-$algebraic varieties $ Z_{\rho }(\sigma )(B)$ in $ B^{n}, $ which are the set of zeros in $ B^{n} $ of the system of polynomial $ \rho-$congruence equations given by $ \sigma . $ When $ \rho $ is a prime congruence, we find these varieties satisfying the axiom of closed sets, and forming a (Zariski) topology on $ B^{n}. $ Some results about their structures including a version of Nullstellensatz of congruences are obtained.

math.RA↗
arXiv2015

On Twists of A Family of Elliptic Curves and Their $ L-$Function

Let $ E $ be an elliptic curve defined over a number field, the conjecture of Birch and Swinnerton-Dyer (BSD, for short) asserts a deep relation between the group $ E(K) $ of rational points and the $ L-$function $ L(E/K, s)$ of $ E $ at $ s = 1. $ Very few explicit results about $ E(K) $ and $ L(1) $ are known, even no general method is known to determine $ L(1) $ vanishing or not for a given elliptic curve. In this paper, we study some quantities related to BSD of a special class of elliptic curves, more precisely, we study the arithmetic of quadratic twists of elliptic curves $ y^{2} = x(x + \varepsilon p )(x + \varepsilon q) $ and their $L-$function. Based on some classical works, especially those of Greenberg, Kramer-Tunnell, Kato-Rohrlich, Manin and Mazur, under some conditions, we obtain results about the vanishing of the value at $ s = 1 $ of the $ L$-function, and explicitly determine the following quantities: the norm index $ \delta (E, \Q, K), $ the root numbers, the set of anomalous prime numbers, a few prime numbers at which the image of Galois representation are surjective. We also study the relation between the ranks of the Mordell-Weil groups, Selmer groups and Shafarevich-Tate groups, and the structure about the $ l^{\infty }-$Selmer groups and the Mordell-Weil groups over $ \Z_{l}-$extension via Iwasawa theory. These results provide some useful evidence toward verifying the BSD for a family of elliptic curves.

math.NT↗
arXiv2015

On cohomology groups $ H^{1} $ of $ G-$modules of finite type over cyclic groups

Let $ G $ be a cyclic group, in this paper, we study the Herbrand quotient and $ 1-$th cohomology group on finitely generated $ G-$modules in some cases. When $ G $ is of order $ 2, $ the order of the cohomology group is explicitly related to some invariants, and this relation is used to study unit groups over quadratic extensions of number fields. We also give some applications on Pell equations and class number of number fields.

math.NT↗
arXiv2014

On quadratic twists of elliptic curves and some applications of a refined version of Yu's formula

In this paper, we study some cohomology groups and quadratic twists of elliptic curves, and apply Tate local duality and the results of Kramer-Tunnell on local norm cokernel to give a refined version of Yu's formula in the case of elliptic curves. Then, by using this refinement formula, we obtain explicit orders of Shafarevich-Tate groups of some elliptic curves in quadratic number fields, including a few unconditional cases.

math.NT↗
arXiv2014

The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces $ X $ and $ Y, $ let $ \mathcal{M}_{\flat } (X), \mathfrak{M}(V \rightarrow W) $ and $ \mathfrak{D}_{\flat }(X, Y) $ be the ballean of $ X $ (the family of the balls in $ X $), the space of mappings from $ X $ to $ Y, $ and the space of mappings from the ballen of $ X $ to $ Y, $ respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., $ \widehat{ρ} _{u}, \widehat{β}_{X, Y}^{λ}, \widehat{β}_{X, Y}^{\ast λ} $) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable $ λ, $ including some normed algebra structure. To some extent, the class $ \widehat{β}_{X, Y}^{λ} $ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when $ X $ is compact and $ Y = K $ is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of $ K-$valued measures on $ X. $

math.MG↗
arXiv2012

Lattice-ordered matrix algebras over real GCD-domains

Let $ R \subset \R $ be a GCD-domain. In this paper, Weinberg's conjecture on the $ n \times n $ matrix algebra $ M_{n}(R) \ (n \geq 2) $ is proved. Moreover, all the lattice orders (up to isomorphisms) on a full $ 2 \times 2 $ matrix algebra over $ R $ are obtained.

math.RA↗
arXiv2010

A note on $ L(1) $ of Hecke $ L-$series associated to the elliptic curves with CM by $ \sqrt{-3} $

Consider elliptic curves $ E:\ y^{2} = x^{3} + D^{3} $ defined over the quadratic field $\ \Q(\sqrt{-3}) $. Hecke $ L-$series attached to $ E $ are studied, formulae for their values at $ s=1, $ and bound of 3-adic valuations of these values are given. These results are complementary to those in [Q] and [QZ], and are consistent with the predictions of the conjecture of Birch and Swinnerton-Dyer.

math.NT↗
arXiv2009

On Some Additive Properties of Multiplicative Subsemigroups of Semirings and Arithmetic Applications I

In this paper, we consider a question of sum-keeping about a multiplicative subsemigroup and its generator subsets in a semiring, and develop some elementary (collapse) process of the sum-keeping retraction through subsets until one minimal generators subset. As an application, we study and analyze several classical problems in additive number theory on the semiring of non-negative integers by this algebraic and combinatory idea, and provide new proofs in more simple and direct way for several classical results in number theory. Some further questions are also presented and discussed.

math.NT↗
arXiv2009

On several families of elliptic curves with arbitrary large Selmer groups

In this paper, we calculate the $ ϕ(\hatϕ)-$Selmer groups $ S^{(ϕ)} (E / \Q) $ and $ S^{(\hatφ)} (E^{\prime} / \Q) $ of elliptic curves $ y^{2} = x (x + εp D) (x + εq D) $ via descent theory (see [S, Chapter X]), in particular, we obtain that the Selmer groups of several families of such elliptic curves can be arbitrary large.

math.AG↗
arXiv2009

On some congruence properties of elliptic curves

In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of congruence properties of elliptic curves are also presented and discussed.

math.NT↗
arXiv2008

Geometry of Non-Archimedean Gromov-Hausdorff distance

In this paper, we study the geometry of non-Archimedean Gromov-Hausdorff metric. This is the first part of our series work, which we try to establish some facts about the counterpart of Gromov-Hausdorff metric in the non-Archimedean spaces. One of the motivation of this work is to find some implied relations between this geometry and number theory via p-adic analysis, so that we can use the former as a tool to study the relating arithmetic aspects.

math.MG↗
arXiv2001

Mordell-Weil groups and Selmer groups of two types of elliptic curves

Consider elliptic curves $ E=E_σ: y^2 = x (x+σp) (x+σq), $ where$ σ=\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(ϕ)}(E/{\mathbf{Q})}$, and $\ S^{(\hatϕ)}(E/{\mathbf{Q})} $ are explicitly determined, e.g., $\ S^{(2)}(E_{+1}/{\mathbf{Q}})= $ $({\mathbf{Z}}/2{\mathbf{Z}})^2; $ $ ({\mathbf{Z}}/2{\mathbf{Z}})^3; $ or $ ({\mathbf{Z}}/2{\mathbf{Z}})^4 $ when $p\equiv 5; 1 $ or $3; $ or $ 7 ({\mathrm{mod}} 8)$ respectively. (2) When $p\equiv 5 (3, 5$ for $σ=-1) ({\mathrm{mod}} 8), $ it is proved that the Mordell-Weil group $ E({\mathbf{Q})} \cong $ $ {\mathbf{Z}}/2{\mathbf{Z}} \oplus{\mathbf{Z}}/2{\mathbf{Z}} $ having rank $0, $ and Shafarevich-Tate group {\CC ':} $(E/{\mathbf{Q}})[2]=0. $ (3) In any case, the sum of rank$E({\mathbf{Q})}$ and dimension of {\CC ':} $(E/{\mathbf{Q}})[2] $ is given, e.g., $0; 1; 2 $ when $p\equiv 5; 1 $ or $3; 7 ({\mathrm{mod}} 8)$ for $σ=1$. (4) The Kodaira symbol, the torsion subgroup $E(K)_{tors}$ for any number field $K$, etc. are also obtained. This paper is a revised version of ANT-0229.

math.NT↗
arXiv2001

L-series and their 2-adic and 3-adic valuations at s=1 attached to CM elliptic curves

$L-$series attached to two classical families of elliptic curves with complex multiplications are studied over number fields, formulae for their special values at $s=1, $ bound of the values, and criterion of reaching the bound are given. Let $ E_1: y^{2}=x^{3}-D_1 x $ be elliptic curves over the Gaussian field $K=\Q(\sqrt{-1}), $ with $ D_1 =π_{1} ... π_{n} $ or $ D_1 =π_{1} ^{2}... π_{r} ^{2} π_{r+1} ... π_{n}$, where $π_{1}, ..., π_{n}$ are distinct primes in $K$. A formula for special values of Hecke $L-$series attached to such curves expressed by Weierstrass $\wp-$function are given; a lower bound of 2-adic valuations of these values of Hecke $L-$series as well as a criterion for reaching these bounds are obtained. Furthermore, let $ E_{2}: y^{2}=x^{3}-2^{4}3^{3}D_2^{2} $ be elliptic curves over the quadratic field $ \Q(\sqrt{-3}) $ with $ D_2 =π_{1} ... π_{n}, $ where $π_{1}, ..., π_{n}$ are distinct primes of $\Q(\sqrt{-3})$, similar results as above but for $3-adic$ valuation are also obtained. These results are consistent with the predictions of the conjecture of Birch and Swinnerton-Dyer, and develop some results in recent literature for more special case and for $2-adic$ valuation.

math.NT↗
arXiv1998

Explicit classification for torsion subgroups of rational points of elliptic curves

The classification of elliptic curves E over the rationals Q is studied according to their torsion subgroups E_{tors}(Q) of rational points. Explicit criteria for the classification are given when E_{tors}(Q) are cyclic groups with even orders. The generator points P of E_{tors}(Q) are also explicitly presented in each case. These results, together with recent results of K. Ono, completely solve the problem of the mentioned explicit classification when E has a rational point of order 2.

math.NT↗