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Jie Shu

Publications and source records attributed to Jie Shu.

11 recordsLinked to original sources
arXiv2025

Quadratic spaces and Selmer groups of abelian varieties with multiplication

For certain symmetric isogeny $\lambda: A\rightarrow A^\vee$ of abelian varieties over a global field $F$, B. Poonen and E. Rains put an orthogonal quadratic structure on $\mathrm{H}^1(\mathbb{A}_F,A[\lambda])$ and realize the Selmer group $\mathrm{Sel}_\lambda(A)$ as an intersection of two maximal isotropic subspaces of $\mathrm{H}^1(\mathbb{A}_F,A[\lambda])$. With this understanding of Selmer groups, they expect to model the Selmer groups of elliptic curves and Jacobian varieties of hyperelliptic curves as the intersections of random maximal isotropic subspaces of orthogonal spaces. We extend this phenomenon to abelian varieties with multiplication and discuss the Shafarevich-Tate groups.

math.NT↗
arXiv2025

Selmer ranks in twists of CM abelian varieties

We prove the Selmer ranks in certain families of $p$-th twists of CM abelian varieties obey the symplectic or unitary distributions. As an application, for a prime $p\geq 3$, we obtain that the twisted Fermat curves $X^p+Y^p=\delta$ over a number field containing a primitive $p$-th root of unity are ``largely" unsolvable as $\delta$ varies. We also discuss the rank growth in cyclic extensions of prime degree for CM abelian varieties.

math.NT↗
arXiv2025

Revealing the internal magnetic field configuration of magnetars via their associated periodic signals

The magnetic deformation of magnetars is affected by their internal magnetic fields, which are generally difficult to be measured directly through observations. In this work, the periodic pulse-phase modulations in the hard X-ray emissions of the magnetars 4U 0142+61, 1E 1547.0-5408, SGR 1900+14, and SGR 1806-20, and the periodicities of fast radio bursts (FRBs) 180916 and 121102 are interpreted as free precession of the (host) magnetars. Using these periodic signals, we investigate the magnetars' internal magnetic fields. In order to simultaneously account for the modulation periods and surface thermal emissions of the former four magnetars, and require that their internal poloidal fields smoothly connect with the surface dipole fields, the parameter that characterizes the distribution of toroidal field in the magnetar interior should satisfy $\beta\gtrsim1$. Moreover, their volume-averaged strengths of poloidal and toroidal fields are respectively $\bar{B}_{\rm p}\sim10^{14}$--$10^{15}$ G and $\bar{B}_{\rm t}\sim10^{15}$ G with the strength ratios $\bar{B}_{\rm t}/\bar{B}_{\rm p}$ generally distributing within $\sim2$--$37$. We could also constrain the critical temperature for neutron superfluidity in the neutron-star core considering that the former four magnetars are probably precessing, and the most stringent constraint is $T_{\rm c,core}<6.4\times10^8$ K. Adopting a possible critical temperature $T_{\rm c,core}=5\times10^8$ K, we could obtain $\bar{B}_{\rm p}\gtrsim10^{14}$--$10^{15}$ G and $\bar{B}_{\rm t}\gtrsim10^{14}$--$10^{15}$ G for the host magnetars of FRBs 180916 and 121102, which indicates that the magnetars of our interest possibly have similar poloidal and toroidal fields.

astro-ph.HE↗
arXiv2021

Root numbers and Selmer groups for the Jacobian varieties of Fermat curves

Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=δ$ where $δ$ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When $p\nmid δ$, we bound the Selmer groups of these Jacobian varieties. Moreover, if $p$ is regular and all prime ideals of $K$ dividing $δ$ are inert in $K/F$, the Selmer groups are explicitly determined and we verify the $p$-parity conjectures of these Jacobian varieties. We also give an asymptotic lower bound for the number of Fermat Jabobians for which the $p$-parity conjecture holds.

math.NT↗
arXiv2021

Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over $\mathbb{Q}$. We prove the existence of explicit infinite families of quadratic twists with analytic ranks $0$ and $1$ for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank $1$. In addition, we show that these families of quadratic twists satisfy the $2$-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

math.NT↗
arXiv2019

Cube sums of form $3p$ and $3p^2$ II

Let $p\equiv 2,5\mod 9$ be a prime. We prove that both $3p$ and $3p^2$ are cube sums. We also establish some explicit Gross-Zagier formulae and investigate the 3 part full BSD conjecture of the related elliptic curves.

math.NT↗
arXiv2019

Waldspurger's period integral for newforms

In this paper we discuss Waldspurger's local period integral for newforms in new cases. The main ingredient is the work \cite{HN18} on Waldspurger's period integral using the minimal vectors, and the explicit relation between the newforms and the minimal vectors. We use a representation theoretical trick to simplify computations for newforms. As an example, we compute the local integral coming from a special arithmetic setting which was used to study 3-part full BSD conjecture in \cite{HSY}.

math.NT↗
arXiv2018

An explicit Gross-Zagier formula related to the Sylvester Conjecture

Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier formula, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.

math.NT↗
arXiv2018

Cube sums of form $3p$ and $3p^2$

Let $p\equiv 2,5\mod 9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross-Zagier formulae for these Heegner points and give variants of the Birch and Swinnerton-Dyer conjecture of the related elliptic curves.

math.NT↗
arXiv2014

Cube Sum Problem and an Explicit Gross-Zagier Formula

A nonzero rational number is called a cube sum if it is of form $a^3+b^3$ with $a,b\in \mathbb{Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We give also a general construction of Heegner point and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curve related to the cube sum problem.

math.NT↗