arXiv subjects

Khai-Hoan Nguyen-Dang

Publications and source records attributed to Khai-Hoan Nguyen-Dang.

18 recordsLinked to original sources

arXiv2026

Rigid analytic 1-motives and conjugate uniformization of abeloid varieties

Let $K$ be a $p$-adic field. We study the arithmetic theory of abeloid varieties over $K$. Our aims are twofold. First, we study the theory of rigid analytic 1-motives, which will be viewed as a tool to describe degeneration of abeloid varieties, similarly as in the classical algebraic setting. Our key new results are the equivalence between formal (resp. log formal) 1-motives over $\mathcal{O}_K$ and rigid analytic 1-motives with good (resp. semi-stable) reduction over $K$, and the Néron-Ogg-Shafarevich criterion for the good (resp. semi-stable) reduction of rigid analytic 1-motives. In particular, we construct log formal 1-motives and log $p$-divisible groups over $\mathcal{O}_K$ from semi-stable abeloid varieties over $K$. Next, we study the conjugate uniformization of an arbitrary abeloid variety $A$ over $K$. This is a type of $p$-adic uniformization initiated by Iovita--Morrow--Zaharescu in case of abelian varieties with good reduction. Our approach here is based on Fargues' theory of $p$-divisible rigid analytic groups. In fact, we view the theory of conjugate uniformization as a study of rational points of dualizable $p$-divisible rigid analytic groups in terms of their classification Hodge--Tate triples. Along the way, we construct $p$-divisible rigid analytic groups from rigid analytic 1-motives.

math.NT↗

arXiv2026

Torsion, Betti rank, and small points in semiabelian schemes

Let $\mathcal G\to S$ be a semiabelian scheme over a smooth complex algebraic base, with abelian relative dimension $g$ and toric rank $r$. We prove that a dominating irreducible subvariety not contained in a proper relative special subvariety has Zariski-dense fibrewise torsion if and only if its mixed Betti rank is $2(g+r)$. In particular, its dimension is at least $g+r$. For every prime, primary torsion is then dense even among points of maximal rank. The exceptional loci include generalized Ribet loci. Over algebraic numbers, we prove that generic small points force full mixed rank whenever the abelian Betti rank is maximal. This yields a transfer from abelian relative Bogomolov dimension theorems to arbitrary toric extensions, together with toric height gaps above abelian torsion and order-linear gluing-height bounds for nodal Pell equations. The proof combines algebraic quotients of marked $1$-motives, quadratic nef compensation, a central-monodromy calculation, and algebraicity of mixed Betti strata.

math.AG↗

arXiv2026

Chern flow and Chern moment algebras

We construct realizable-volume models over every field for the factorial normalizations of homogeneous Lascoux, Lascoux-atom, and positive Grothendieck packets, including their minimal homogenizations and layers. In particular, the construction realizes the factorial normalizations of all Schubert and key polynomials and of the minimal sign-corrected homogeneous Grothendieck polynomials. The normalized polynomials are Lorentzian, and the ordinary supports are the lattice points of integral generalized polymatroids. On a Bott--Samelson tower, row and co-row filtrations assemble the local factors into globally generated bundles; a creation-state graph absorbs the remaining kernel factors by Chern flow. We also construct intrinsic algebras of joint Chern moments. Positive inverse-Chern presentations give these algebras Hard Lefschetz and Hodge--Riemann relations, and supply source-level Hodge completions of the packets. For globally generated tropical toric bundles in the sense of Kaveh--Manon, finite generating witnesses and matroid duality provide the presentations required by Larson--Partida's theorem, without representability. These constructions yield joint Chern-number inequalities, nonvanishing polymatroids, and equality criteria.

math.CO↗

arXiv2026

Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties

Silverman conjectured that the geometric divisibility sequence attached to a Zariski-dense point on an irreducible commutative algebraic group of dimension at least two, with no unipotent part, returns to its initial value infinitely often. We construct, for the first time, unconditional examples of this phenomenon on geometrically nonsplit semiabelian varieties over number fields using torsion translates of normalized Ribet points. More precisely, let $A/K$ be a positive-dimensional abelian variety over number field $K$, let \[ 1\longrightarrow\mathbf G_m\xrightarrowιG_q\xrightarrowπA \longrightarrow0 \] be the extension represented by $q\in A^\vee(K)$, and let $R_β(q)\in G_q(K)$ be the normalized Ribet point associated with a homomorphism $β:A^\vee\to A$. We set $δ:=β-\widehatβ$ and assume that $δ$ is an isogeny and that $\mathbf Z(δq)$ is Zariski dense in $A$. For a torsion point $t\in\mathbf G_m(K)_{\mathrm{tors}}$, identify $t$ with $ι(t)$ and set $P=R_β(q)+t$. Then $P$ has Zariski-dense cyclic orbit in the geometrically nonsplit extension $G_q$. There exists an explicit integer $N_{δ,t}$ such that if $N_{δ,t}>1$, then $N_{δ,t}\mid \mathrm{ord}(\overline P_v)$ at all but finitely many places $v$. Consequently, there is a squarefree integer $Q_P>1$ such that \[ (n,Q_P)=1 \quad\Longrightarrow\quad \mathfrak d_{\mathcal N}(nP)=\mathfrak d_{\mathcal N}(P), \] where $\mathfrak d_{\mathcal N}$ denotes the full denominator ideal on the Néron lft-model $\mathcal N$. In particular, we construct explicitly a geometrically nonsplit semiabelian surface $G/\mathbf Q$ and a semiabelian threefold over $\mathbf Q$ satisfying the Silverman conjecture. It follows that we can construct instances satisfying the Silverman conjecture for every dimension at least two.

math.NT↗

arXiv2026

The minimal periodicity for integral bases of pure number fields

Fix $n\ge3$. For the pure field $K_a=\Bbb Q(θ)$ with $θ^n=a$, where $a\neq \pm1$ is $n$th-power-free, $x^n-a$ is irreducible over $\Bbb Q$, and Hypothesis~\textup{(\textbf{H})} holds, we encode an integral basis in the fixed coordinate $\{1,θ,\dots,θ^{n-1}\}$ by its \emph{shape}. We prove a sharp local-to-global principle: for each $p^e\!\parallel n$, the local shape at $p$ is determined by $a\bmod p^{\,e+1}$, and this precision is optimal. Moreover, the global shape is periodic with minimal modulus \[ M(n)=\prod_{p^e\parallel n}p^{\,e+1}=n\cdot\mathrm{rad}(n), \] providing many applications in the understanding integral bases of pure number fields.

math.NT↗

arXiv2026

On the sequence $\mathrm{gcd}(a^n-1,b^n-1)$

For integers $a,b\ge 2$, put \[ g_n:=\gcd(a^n-1,b^n-1)\qquad(n\ge 1). \] We prove that $(g_n)$ satisfies a constant-coefficient linear recurrence if and only if $a$ and $b$ are multiplicatively dependent; the same conclusion holds if only a tail of $(g_n)$ is assumed to satisfy such a recurrence. More generally, if $a$ and $b$ are multiplicatively independent, then for every fixed shift $N\ge 0$, every integer sequence $(W_n)_{n\ge 1}$ that satisfies a constant-coefficient linear recurrence and the pointwise divisibilities \[ W_n\mid a^{n+N}-1 \qquad\text{and}\qquad W_n\mid b^{n+N}-1 \qquad(n\ge 1) \] is eventually periodic. If $(W_n)$ satisfies a recurrence with nonzero trailing coefficient, then it is periodic from the first term. In particular, every common integer linear divisibility-sequence factor of $a^n-1$ and $b^n-1$, in the convention adopted here, is periodic. The proof combines the Bugeaud--Corvaja--Zannier bound with the structure of subexponential linear recurrences and a local valuation argument.

math.NT↗

arXiv2026

Richardson volume models for skew Schur and skew Schur $P/Q$-functions

We identify ordinary skew Schur polynomials and skew Schur $P$-functions as top-degree total-Chern intersection polynomials on Richardson varieties in ordinary and Lagrangian Grassmannians. We then obtain that \[ \mathcal N(s_{λ/μ}),\qquad \mathcal N(P_{λ/μ}),\qquad \mathcal N(Q_{λ/μ}) \] are realizable volume polynomials. This settles the skew-Schur and Schur-$P$ Lorentzian conjectures of Huh--Matherne--Mészáros--St.~Dizier and strengthens the latter to arbitrary skew $P/Q$-functions. The constructions extend to cycle transforms attached to arbitrary irreducible subvarieties of ordinary and Lagrangian Grassmannians. Their realizable-volume interpretation yields reverse Khovanskii--Teissier and Lorentzian Hodge--Riemann inequalities for ordinary and shifted tableau multiplicities; exact ordinary skew-Schur support permutahedra and extremal coefficients; the known straight shifted support polytopes with their vertex coefficients; implicit exact permutahedra for arbitrary skew $P/Q$-functions; and weighted-aggregation, covariance, and two-row Littlewood--Richardson consequences. An appendix by Zhenpeng Wang constructs the dual type~$D$ spinor cycle transform and a direct Richardson realization of $Q_{λ/μ}$. In characteristic two, compatible very special isogenies induce finite flat radicial morphisms between the ambient Lagrangian and spinor models. Their restrictions to the corresponding Richardson varieties have degrees $2^{\ell(λ)-\ell(μ)}$ and $2^{|λ|-|μ|-\ell(λ)+\ell(μ)}$, and a regular complete-intersection bridge explains the difference between the two projective-bundle shifts.

math.CO↗

arXiv2026

$α$-monogeneity of pure number fields: criterion and density

Let $n\ge 2$, let $m\in\mathbb Z\setminus\{0\}$, and let $K=\mathbb Q(α)$, where $α^n=m$ and $X^n-m$ is irreducible over $\mathbb Q$. We study when the natural order $\mathbb Z[α]$ is the full ring of integers $\mathcal O_K$. For the pure family $X^n-m$, we give a short proof, using only Dedekind's index criterion, of the equivalence $\mathcal O_K = \mathbb Z[α]$ iff $m$ is square-free and $ν\_p(m^p-m)=1$ for every prime $p\mid n$. Equivalently, the prime support of $[\mathcal O_K:\mathbb Z[α]]$ is $$\{p:ν_p(m)\ge 2\}\cup \{p\mid n:ν_p(m^p-m)\ge 2\}.$$ We then compute the natural density of the corresponding parameters in the one-parameter family $X^n-m$: $$δ_n=\frac{6}{π^2}\prod_{p\mid n}\frac{p}{p+1}.$$ We also give an arithmetic-progression refinement, a density-theoretic independence statement for the local obstruction sets at primes dividing $n$, and discriminant-ordered counts of the corresponding fields.

math.NT↗

arXiv2026

Counterexamples of Friedlander--Iwaniec dual sums conjecture

Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\sum_{n\ge1}a(n)n^{-s}, \qquad B(s)=\sum_{n\ge1}b(n)n^{-s},$$ be the two Dirichlet series related by a certain functional equation. Let $m$ be the \emph{analytic degree} of the functional equation. For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\mathcal B_{\ell,D}(x,N) := \sum_{\substack{n\in\mathbb N\\ n\le N}} b(n)n^{-β_m} \cos\left( 2πm\left(\frac{nx}{D}\right)^{1/m} +\frac{π\ell}{4} \right),$$ where $D\ge1$ is the conductor, $β_m:=\frac{m+1}{2m}$, and $\ell=m-3-2k$ is determined by the archimedean weight $k$ of the functional equation. Their Conjecture 1 predicts that, for every $\varepsilon>0$, $$\mathcal B_{\ell,D}(x,N) \ll_{\varepsilon,\boldsymbolκ} (DNx)^\varepsilon,$$ uniformly in the variables $x$ and $N$, with the degree, conductor, and archimedean datum fixed. We give counterexamples to this prediction with $$A(s)=B(s)=ζ(s)^m,\; m\geq 4$$ where $$ζ(s):=\sum_{n\ge1}n^{-s} \qquad(\operatorname{Re}s>1)$$ is the Riemann zeta function.

math.NT↗

arXiv2026

Remarks on a theorem of Silverman

Motivated by a theorem of Silverman, we consider the following problem. Let $A$ be an abelian variety over a global field $K$. Given a non-torsion point $P \in A(K)$, for a sufficiently large positive integer $n$, whether there exists a place $v$ of $K$ such that the order of the reduction of $P$ modulo $v$ is $n$? In this article, we first show that this holds for an elliptic curve over a global function field of positive characteristic $p>3$ and for sufficiently large positive integers $n$ coprime to $p$. In the second part of the paper, we consider its relative version over $\mathbb C$. More precisely, let $π: \mathcal{A} \rightarrow S$ be an abelian scheme over some variety $S$ over $\mathbb C$, and let $P$ be a non-torsion section of $π$. If the Betti map associated to $P$ is generically submersive, then for every sufficiently large $n$, there is a point $s$ in $S(\mathbb C)$ such that $P(s)$ is a point of order $n$ in the corresponding fiber.

math.NT↗

arXiv2026

Impossibility of a nontrivial Brunn--Minkowski inequality for higher Dirichlet eigenvalue

Let $λ_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that $λ_1$ satisfies a Brunn--Minkowski inequality: $K \mapsto λ_1(K)^{-1/2}$ is concave on the family of convex bodies. We show that no analogous statement holds for higher eigenvalues. More precisely, for any $j \geq 2$ and $N \geq 2$, if $K \mapsto (f \circ λ_j)(K)$ is concave on the family of convex bodies in $\mathbb{R}^N$ for some function $f: (0, \infty) \to \mathbb{R}$, then $f$ must be constant.

math.MG↗

arXiv2026

Minkowski shapes of pure number fields

We study the Minkowski shape of pure number fields \[ K_a=\mathbb Q(θ),\qquad θ^n=a. \] For admissible parameters satisfying an explicit local hypothesis at the primes dividing $n$, we prove a discrete--archimedean factorization \[ \mathrm{sh}(K_a)=\bigl[C(a)^{\mathsf T}\mathrm{diag}\bigl(s_1(a),\dots,s_{n-1}(a)\bigr)C(a)\bigr], \] where the $s_m(a)$ arise from normalized monomials and $C(a)\in\mathrm{GL}_{n-1}(\mathbb Q)$ comes from a normalized integral basis. This yields a uniform odd/even rigidity dichotomy: for every odd $n\geq 3$, the Minkowski shape is a complete invariant among admissible pure degree-$n$ fields, whereas for $n=2r$ it determines the core field $\mathbb Q(|a|^{1/r})$; on the squarefree admissible subfamily it is complete up to sign, although infinitely many non-isomorphic pairs $K_a$ and $K_{-a}$ have the same shape. We also derive explicit formulas for $|\mathrm{disc}(K_a)|$, including exponent-vector and divisor-lattice factorizations. Finally, we show that the pure-field shape locus is supported on rational diagonal leaves in shape space: unconditionally it lies in a countable union of closed leaves, while under the same local hypothesis only finitely many leaves occur in each fixed degree. On a fixed normalized stratum, the shape depends only on ratio variables, whereas discriminant growth is governed by independent product variables.

math.NT↗

arXiv2026

Legendre compressions and an integrality conjecture for the Hörmander--Bernhardsson extremal function

We prove Conjecture~2 of Bondarenko, Ortega-Cerdà, Radchenko, and Seip for the three-term recurrence attached to the Hörmander--Bernhardsson extremal function $φ$. More precisely, define \[ \widetilde u_{-1}=0,\qquad \widetilde u_0=1, \] and \[ \widetilde u_{n+1} = \frac{4n+2}{n+1}\bigl(n(n+1)-λ\bigr)\widetilde u_n + \frac{4n}{n+1}x\,\widetilde u_{n-1}. \] Then \[ \widetilde u_n(x,λ)\in\mathbb Z[x,λ] \qquad(n\ge0). \] The proof is a determinant comparison in the scaled Legendre basis. After sign reversal and central-binomial normalization, the recurrence becomes exactly the continuant recurrence of a finite tridiagonal compression. In particular, if $T_n(a,λ)$ denotes the $n$th BOCRS tridiagonal truncation, then \[ \widetilde u_{n+1}(a^2,λ)=\binom{2n+2}{n+1}\det T_n(a,λ). \] As consequences, we derive that \[ \left(\fracπ{4C}\right)^2 \quad\text{and}\quad -\frac{L_τ(1)}{2C} \] are not simultaneously rational, where $C$ is the sharp point-evaluation constant for $PW^1$, $\pmτ_n$ are the nonzero zeros of $φ$, and $ L_τ(1)=\sum_{n\ge1}\frac{(-1)^n}{τ_n}.$ Finally, if we write $φ(z)=\sum_{n\ge0}c_n z^{2n},$ then \[ c_n\in C^n\,\mathbb Z[π^2,C,L_τ(1)] \qquad(n\ge0). \]

math.NT↗

arXiv2026

Kummer-Artin-Schreier-Witt Theory

We study the problem of lifting the Artin--Schreier--Witt isogeny from characteristic $p>0$ to characteristic $0$, which is central to the lifting problem for Galois covers of algebraic schemes in positive characteristic. We introduce a new technique that associates a Kummer class, representing a tamely ramified cyclic extension, to a Witt vector via Matsuda's Kummer--Artin--Schreier--Witt theory. This viewpoint leads to an explicit construction of a lift of the isogeny over a concrete base ring. Our results lay the groundwork for further applications, including the study of inseparable extensions and Kato's refined Swan conductor.

math.NT↗

arXiv2026

Eisenstein-prime Obstruction Sieve for Monogenicity

Alpöge--Bhargava--Shnidman showed that even a strengthened \emph{no local obstruction} condition for monogenicity does not force a global power integral basis: in the full spaces of cubic and quartic fields, a positive proportion are non-monogenic yet satisfy this ABS fixed-sign condition. This raises a natural family-level question: does the same phenomenon persist inside one-parameter families, where the local structure varies in a highly constrained way? In this paper we answer this in the negative for the pure fields $K_m=\mathbb Q(α)$ with $α^n=m$ ($n\ge 4$) and $m$ square-free. Writing $g(m)=[\mathcal O_{K_m}:\mathbb Z[α]]$, we prove that the set of square-free $m$ for which $g(m)>1$ but $K_m$ has no ABS local obstruction has natural density $0$. Consequently, in the pure family monogenicity and $α$--monogenicity have the same natural density. The proof isolates a reusable mechanism, which we call the Eisenstein-prime obstruction sieve. The argument is packaged in an abstract template and transfers to other Eisenstein parameter families.

math.NT↗

arXiv2026

Admissibility of Hörmander--Bernhardsson extremal zeros

Let $φ$ be the Hörmander--Bernhardsson extremal function, and let $(\pmτ_n)_{n\ge1}$ be its real zeros. Using the recent analytic description of the zero set ${τ_n}$, we prove that the squared zeros $λ_n=τ_n^{2}$ form an admissible sequence in the sense of Quine--Heydari--Song: the heat trace $Θ(t)=\sum_{n\ge1}e^{-λ_n t}$ has a full $t\to0^{+}$ expansion in pure powers of $t^{1/2}$. The proof is based on an analytic normal form \[ λ_n=\Bigl(n+\tfrac12\Bigr)^2+q!\Bigl(\Bigl(n+\tfrac12\Bigr)^{-2}\Bigr), \] a uniform Taylor expansion in $t$, and a Mellin--Hurwitz zeta analysis of the resulting weighted Gaussian sums. As applications we obtain meromorphic continuation and special-value information for the associated spectral zeta function and zeta-regularized product, sharp large-parameter asymptotics for the canonical product $\prod_{n}(1+z/λ_n)$. In particular, we deduce the conjecture by Bondarenko--Ortega-Cerdà--Radchenko--Seip for the special values of the Dirichlet-type series attached to $φ$. We also establish a parity dichotomy: sequences $(τ_n^m)$ are QHS--admissible for even $m$, while for odd $m$ a nonzero $t\log t$ term obstructs admissibility.

math.CV↗

arXiv2025

Computing intrinsic volumes of sublevel sets and applications

Intrinsic volumes are fundamental geometric invariants generalizing volume, surface area, and mean width for convex bodies. We establish a unified Laplace-Grassmannian representation for intrinsic and dual volumes of convex polynomial sublevel sets. More precisely, let $f$ be a convex $d$-homogeneous polynomial of even degree $d \ge 2$ which is positive except at the origin. We show that the intrinsic and dual volumes of the sublevel set $[f \le 1]$ admit Laplace-type integral formulas obtained by averaging the infimal projection and restriction of $f$ over the Grassmannian. This explicit representation yields three main consequences: (1) Löwner--John-type existence and uniqueness results extending beyond the classical volume case; (2) a block decomposition principle describing factorization of intrinsic volumes under direct-sum splitting; (3) a coordinate-free formulation of Lipschitz-type lattice discrepancy bounds. These formulas enable analytic treatment of a broad class of geometric quantities, providing direct access to variational and arithmetic applications as well as new structural insights.

math.MG↗

arXiv2025

Supersymmetric Schur polynomials have saturated Newton polytopes

We prove that every supersymmetric Schur polynomial has a saturated Newton polytope (SNP). Our approach begins with a tableau-theoretic description of the support, which we encode as a polyhedron with a totally unimodular constraint matrix. The integrality of this polyhedron follows from the Hoffman-Kruskal criterion, thereby establishing the SNP property.

math.CO↗