Searcharxiv⌕ Search

arXiv subjects

Chang Lv

Publications and source records attributed to Chang Lv.

At least 19 recordsLinked to original sources
arXiv2026

Descent and Brauer-Manin Obstructions on Deligne-Mumford Stacks

We generalize and compare local-global obstructions for algebraic stacks over number fields. For smooth separated Deligne-Mumford stacks of finite type with a quasi-projective coarse moduli space, we prove that the descent obstruction is contained in the Brauer--Manin obstruction. By lifting $\mathbb{G}_m$-gerbes, we obtain an inclusion between the corresponding composite obstructions. We also show that in this setting the descent obstruction coincides with both the \'etale Brauer-Manin obstruction and the iterated descent obstruction. The Brauer-Manin obstruction also coincides with the iterated Brauer-Manin obstruction.

math.AG↗
arXiv2026

Comparing local-global obstructions on algebraic stacks

Let k be a number field. For quotient stacks [X/G] with X a smooth quasi-projective geometrically integral k-variety and G a linear k-group, we extend several relations between local-global obstructions previously known for varieties, showing that \'etale-Brauer obstruction is the finest among various obstructions (such as iterated descent, finite descent, descent, Brauer-Manin).

math.AG↗
arXiv2026

Cohomological descent for obstructions to local-global principle

We develop a formalism of cohomological descent encoding adelic points and obstructions to local-global principle on algebraic stacks. As an application, by constructing new obstructions using the formalism, we obtain some comparison results of obstructions on some classes of algebraic stacks.

math.AG↗
arXiv2024

Iterated descent obstructions for algebraic stacks

Base on a conjecture, we prove that for any smooth separated stack of finite type over a number field, its descent obstruction equals its iterated descent obstruction. As a consequence, we show that for any algebraic stack over a number field that has a finite etale covering of a smooth geometrically integral variety, its descent obstruction equals its iterated descent obstruction.

math.NT↗
arXiv2023

The Brauer-Manin obstruction on algebraic stacks

For algebraic stacks over number fields, we define their Brauer-Manin sets, Brauer-Manin pairings, and extend the descent theory of Colliot-Th\'el\`ene and Sansuc. By extending Sansuc's exact sequence, we show the torsionness of Brauer groups of stacks that are locally quotients of varieties by linear groups. With mild assumptions, for stacks that are locally quotients or Deligne-Mumford, we show that the Brauer-Manin obstruction coincides with some other cohomological obstructions such as obstructions given by torsors under connected groups or abelian gerbes. For Brauer-Manin sets of these stacks, we show the properties such as descent along a torsor, product preservation are still correct. These results extend classical theories of those on varieties.

math.AG↗
arXiv2021

On commutative diagrams consisting of low term exact sequences

We establish several useful commutative diagrams consisting of low term exact sequences attached to {\Grot} spectral sequences, which extends and integrates the previous ones appeared in literature such as Alexei~N. Skorobogatov [Beyond the {M}anin obstruction, Invent. Math. (1999)], and [On the elementary obstruction to the existence of rational points, Mathematical Notes (2007)]. Parts of the diagrams was frequently used in local-global principle to rational points.

math.AG↗
arXiv2021

Constant Tamagawa numbers of special elliptic curves

For the elliptic curves $E_{\sigma 2D} : y^2 = x^3 + \sigma 2Dx$ , which has 2-isogeny curve $E'_{\sigma 2D} : y^2 = x^3 -\sigma 8Dx$, $\sigma = \pm 1,\ D = p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}$, where $p_i$ are different odd prime numbers and $e_i = 1 \text{ or } 3$, we demonstrate that Tamagawa numbers of these elliptic curves are always one or zero by the use of matrix in finite field $\mathbb F_2$. The specific number depends on the value of $\sigma$. By our proofs of these results, we find a method to quickly sieve a part of the elliptic curves with Mordell-Weil rank zero or rank one in this form as an application.

math.NT↗
arXiv2020

The second descent obstruction and gerbes

We extend the notion of rational points and cohomological obstructions on varieties to categories fibred in groupoids. We also establish the generalized theory of descent by torsors. Then we interpret the obstruction given by the second cohomology of abelian sheaves in terms of categorical points of gerbes, analogue to descent by torsors. As an application, we construct some composite obstructions not larger than the descent obstruction. We also propose some new kinds of obstructions including the derived obstruction, which has good behavior under a product and is also not larger than the descent obstruction.

math.AG↗
arXiv2018

On Ring Class Fields of Number Rings

For a number field $K$, we extend the notion of the ring class field of an order in $K$ [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary number ring in $K$. We give both ideal-theoretic and idele-theoretic description of this number ring class field, and characterize it as a subfield of the ring class field of some order. As an application, we use it to give a criterion of the solvability of a higher degree norm form equation over a number ring and finally describe algorithms to compute this field.

math.NT↗
arXiv2018

Nonexistence of generalized bent functions and the quadratic norm form equations

We present a new result on the nonexistence of generalized bent functions (GBFs)from (Z/tZ)^n to Z/tZ (called type [n, t]) for a large class. Assume p is an odd prime number. By showing certain quadratic norm form equations having no integral points, we obtain a universalresult on the nonexistence of GBFs with type [n,2p^e] when p and n satisfy a certain inequality, and by computational methods with a widely accepted hypothesis, Generalized Riemann Hypothesis, we also achieve some results on the nonexistence of GBFs for relatively small p.

cs.IT↗
arXiv2015

On the Integral Representation of Binary Quadratic Forms and the Artin Condition

For diophantine equations of the form ax^2+bxy+cy^2+g=0 over Z whose coefficients satisfy some assumptions, we show that a condition with respect to Artin reciprocity map, which we call the Artin condition, is the only obstruction to the local-global principle for integral solutions of the equation. Some concrete examples are presented.

math.NT↗
arXiv2015

On the Integral Representation of $ax^2+by^2$ and the Artin Condition

Given a number field $F$ with $ø_F$ its ring of integers. For certain $a,b$ and $α$ in $ø_F$, we show that the Artin condition is the only obstruction to the local-global principle for integral solutions of equation $ax^2+by^2=α$. Some concrete examples are presented at last.

math.NT↗