arXiv subjects

Li-Tong Deng

Publications and source records attributed to Li-Tong Deng.

5 recordsLinked to original sources

arXiv2026

Elliptic curves with rank one and nontrivial 2-part of Tate Shafarevich groups over the $\mathbb{Z}_2$-extension of $\mathbb{Q}$

Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_2$-extension over $\mathbb{Q}$. For each integer $n\geq1$, let $\mathbb{Q}_n$ denote the unique subfield in $\mathbb{Q}_\infty$ such that $[\mathbb{Q}_\infty:\mathbb{Q}]=2^n$. Denote by $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$ the group ring of ${\rm Gal}(\mathbb{Q}_\infty/\mathbb{Q})$. For any elliptic curve defined over $\mathbb{Q}$ with odd conductor, the Mazur-Tate modular element associated with the curve is an element of $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$. In this paper, for each $n$, we study the $2$-adic properties of Mazur-Tate modular elements associated with quadratic twists of elliptic curves, under specializations by finite order characters of ${\rm Gal}(\mathbb{Q}_n/\mathbb{Q})$. Using the congruence properties of Heegner points and an equivariant version of the Coates-Wiles theorem, we construct an elliptic curve $E/\mathbb{Q}$ and a family of quadratic twists $E^{(m)}$ of $E$ such that each $E^{(m)}$ has both analytic and algebraic rank one over $\mathbb{Q}_\infty$, and whose Tate-Shafarevich group is infinite over $\mathbb{Q}_\infty$.

math.NT↗

arXiv2026

Iwasawa Invariants of Even $K$-groups of Rings of Integers in the $\mathbb{Z}_2$-extension over Real Quadratic Number Fields

Let $F$ be a real quadratic number field, and let $F_{cyc}$ denote its cyclotomic $\mathbb{Z}_2$-extension. For each integer $n\geq0$, let $F_n$ be the unique intermediate field in $F_{cyc}$ such that $[F_n:F]=2^n$. By studying the $2$-adic divisibility of Dirichlet $L$-series at negative integers, we derive an asymptotic formula that determines the order of the $2$-primary part of even $K$-groups of rings of integers of $F_n$ for sufficiently large $n$. As a corollary, we determine their $\lambda$ and $\mu$ invariants. We also establish a lower bound for $n$ beyond which this asymptotic formula holds. Our results have two main applications: (1) For $K=\mathbb{Q}$, $\mathbb{Q}(\sqrt{p})$ or $\mathbb{Q}(\sqrt{2p})$ with $p\equiv\pm3\mod 8$, we determine the structure of the $2$-primary tame kernels $K_2\mathcal{O}_{K_n}(2)$; (2) We explicitly determine the three Iwasawa invariants $\lambda,\mu,\nu$ for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors.

math.NT↗

arXiv2024

Non-commutative Iwasawa theory of abelian varieties over global function fields

Let $A$ be an abelian variety defined over a global function field $F$, and let $p$ be a prime distinct from the characteristic of $F$. Let $F_\infty$ be a $p$-adic Lie extension of $F$ that contains the cyclotomic $\mathbb{Z}_p$-extension $F^{\mathrm{cyc}}$ of $F$. In this paper, we investigate the structure of the $p$-primary Selmer group $\mathrm{Sel}(A/F_\infty)$ of $A$ over $F_\infty$. We prove the $\mathfrak{M}_H(G)$-conjecture for $A/F_\infty$. Furthermore, we show that both the $\mu$-invariant of the Pontryagin dual of the Selmer group $\mathrm{Sel}(A/F^\mathrm{cyc})$ and the generalised $\mu$-invariant of the Pontryagin dual of the Selmer group $\mathrm{Sel}(A/F_\infty)$ are zero, therby proving Mazur's conjecture for $A/F$. We then relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of $A$ over the base field $F$. Assuming the finiteness of the Tate-Shafarevich group, we establish that this corank equals the order of vanishing of the $L$-function of $A/F$ at $s=1$. Finally, we extend a theorem of Sechi - originally proved for elliptic curves without complex multiplication - to abelian varieties over global function fields. This is achieved by adapting the notion of generalised Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalised Euler characteristic of $\mathrm{Sel}(A/F_\infty)$ to the Euler characteristic of $\mathrm{Sel}(A/F^{\mathrm{cyc}})$.

math.NT↗

arXiv2023

Quadratic forms, $K$-groups and $L$-values of elliptic curves

Let $f$ be a positive definite integral quadratic form in $d$ variables. In the present paper, we establish a direct link between the genus representation number of $f$ and the order of higher even $K$-groups of the ring of integers of real quadratic fields, provided $f$ is diagonal and $d \equiv 1 \mod 4$, by applying the Siegel mass formula. When $d=3$, we derive an explicit formula of $r_f(n)$ in terms of the class number of the corresponding imaginary quadratic field and the central algebraic values of $L$-functions of quadratic twists of elliptic curves, by exploring a theorem of Waldspurger. Moreover, by the $2$-divisibility results on the algebraic $L$-values of quadratic twist of elliptic curves, we obtain a lower bound for the $2$-adic valuation of $r_f(n)$ for some odd integer $n$. The numerical results show our lower bound is optimal for certain cases. We also apply our main result to the quadratic form $f=x_1^2+\cdots+x^2_d$ to determine the order of the higher $K$-groups numerically.

math.NT↗

arXiv2023

An application of Birch-Tate formula to tame kernels of real quadratic number fields

Let $F$ be a real quadratic number field with discriminant $D$ and $\mathcal{O}_F$ the ring of integers in $F$. Let $\chi_F$ be the Dirichlet character associated to $F/\mathbb{Q}$. Write $L(\chi_F,s)$ for the Dirichlet L-function of $\chi_F$. By an induction argument for imprimitive Dirichlet L-values, we get several $2$-divisibility results on $L(\chi_F,-1)$ when $D$ has arbitrarily finitely many prime divisors. As an application, by making use of the Birch-Tate formula for $F$, we determine the $2$-primary part for the second $K$ group $K_2\mathcal{O}_F$. We also give a new proof for an old theorem of Browkin and Schinzel.

math.NT↗