Searcharxiv⌕ Search

arXiv subjects

Hui June Zhu

Publications and source records attributed to Hui June Zhu.

At least 19 recordsLinked to original sources
arXiv2026

Generically Ordinary One-parameter Hyperelliptic Families are Dense

Let d=2g+1>4 and let A^{d-1} be the coefficient space parametrizing hyperelliptic families C_a: y^2 = x^d + a_{d-1}x^{d-1} + ... + a_1x+t over the t-line. We show that for a Zariski-generic coefficient vector a=(a_1,...,a_{d-1}) in \bar{Z}^{d-1}, for every prime p large enough, C_a has generically ordinary reduction at every prime above p. For arbitrary a in \bar{Z}^{d-1}, the set of ordinary primes of the hyperelliptic curve C: y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t over Q(a)(t) has natural density at least 1/[Q(a,ζ_d):Q(a)]; after base change to Q(a,ζ_d)(t), it has natural density 1.

math.NT↗
arXiv2026

Construction of Generically Ordinary Families of Hyperelliptic Curves

Katz conjectured in a 2018 lecture that the family of curves $y^2=x^d-dx+t$ over the $t$-line is generically ordinary for all sufficiently large primes $p$. We prove that, for every $g\ge 2$ and every nonzero algebraic integer $α$, the genus-$g$ families $C_α: y^2=x^d+αx+t$ where $d\in\{2g+1, 2g+2\}$ are generically ordinary at every prime $p>P^+(d)$, provided that $α$ is nonzero modulo every prime above $p$. The bound $P^+(d)=d^2-4d+2$ if $d$ is odd, and $P^+(d)=(d^2-3d+2)/2$ if $d$ is even. Finally, we show that for every algebraic integer $α$ and every prime $p\equiv 1 \pmod d$, the family $C_α$ is generically ordinary at every prime above $p$.

math.NT↗
arXiv2026

Construction of Curves with a Controlled First Slope using p-Symmetric Numbers

This paper establishes a constructive link between the first slope of Artin-Schreier curves X_f: y^p-y=f(x) and the p-adic weight of the support of f(x). If the maximal p-adic weight element v in Supp(f) is unique, we show that the first slope's lower bound of 1/s_p(v) is achieved if and only if v satisfies a combinatorial p-adic condition, which we define as p-symmetry. As an application, we construct explicit families of curves in every characteristic p with first slope equal to 1/n for every n>2.

math.AG↗
arXiv2025

Asymptotic Variation of Elementary Abelian p-Extensions over $P^1$

Let A^d denote the coefficient space of all degree-d polynomials f in one variable for some d\ge 3. For any \bar{f} in A^d(\bar\F_p), a rank-\ell Artin-Schreier curve X_{\bar{f},\ell}: y^{p^\ell}-y= \bar{f} is called ordinary if its normalized Newton polygon achieves the infimum in A^d(\bar\F_p). Given \ell and a number field K, we show that there exists a Zariski dense open subset U in A^d, defined over Q, such that if f in U(K) then X_{(f\bmod \wp),\ell} is ordinary for all primes $\wp|p$ with deg(\wp) in \ell\Z and p large enough.

math.NT↗
arXiv2017

On slopes of $L$-functions of $\mathbb{Z}_p$-covers over the projective line

Let $\mathcal{P}: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point, and is unramified at other points. In this paper, we study the $q$-adic valuations of the reciprocal roots in $\mathbb{C}_p$ of $L$-functions associated to characters of the Galois group of $\mathcal{P}$. We show that for all covers $\mathcal{P}$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the valuations of these reciprocal roots are uniformly distributed in the interval $[0,1]$. Furthermore, we show that for a large class of such covers $\mathcal{P}$, the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.

math.NT↗
arXiv2016

Generic Newton Slopes for Artin-Schreier-Witt Tower in two variables

We prove that for any generic polynomial $f$ in two variables of degree $(d_1,d_2)$ over the rationals, for $p$ large enough the Newton slopes of the character power series $C_f^*(χ_m,s)$ of $f$ at $p$ is independent of the choice of the character $χ_m$ (of conductor $p^m$); and the Newton slopes of the $L$-function $L_f^*(χ_m,s)$ of $f$ at $p$ is in weighted arithmetic progression.

math.NT↗
arXiv2016

p-adic variation of L-functions of exponential sums, I

For a polynomial $f(x)$ in $(\mathbb{Z}_p\cap \mathbb{Q})[x]$ of degree $d>2$ let $L(f \bmod p;T)$ be the $L$-function of the exponential sum of $f \bmod p$. Let $\mathrm{NP}(f \bmod p)$ denote the Newton polygon of $L(f \bmod p;T)$. Let $\mathrm{HP}(f)$ denote the Hodge polygon of $f$, which is the lower convex hull in the real plane of the points $(n,n(n+1)/(2d))$ for $0\leq n\leq d-1$. We prove that there is a Zariski dense subset $\mathcal{U}$ defined over $\mathbb{Q}$ in the space $\mathbb{A}^d$ of degree-$d$ monic polynomials over $\mathbb{Q}$ such that for all $f$ in $\mathcal{U}(\mathbb{Q})$ we have $\lim_{p\rightarrow\infty} \mathrm{NP}(f \bmod p) = \mathrm{HP}(f)$. Moreover, we determine the $p$-adic valuation of every coefficient of $L(f \bmod p;T)$ for $p$ large enough and $f$ in $\mathcal{U}(\mathbb{Q})$, and that of $L(x^d+a x \bmod p;T)$ for all $a\neq 0$.

math.AG↗
arXiv2015

On a theorem of Ax and Katz

The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of V. It was strengthened by Adolphson-Sperber in terms of Newton polytope of the support set G of V. In this paper we prove that for every generic algebraic variety over a number field supported on G the Adolphson-Sperber bound can be achieved on special fibre at p for a set of prime p of positive density in SpecZ. Moreover we show that if G has certain combinatorial conditional number nonzero then the above bound is achieved at special fiber at p for all but finitely many primes p.

math.NT↗
arXiv2013

Generic A-family of exponential sums

In this paper we construct a generating polynomial over the rationals for the generic Newton polygon for the L function of exponential sums of the family of f = x^d+ a x^s parameterized by a, and prove some of its key properties. The generating polynomial encodes information of and determines the generic Newton polygon at each prime p when p is large enough, and vice versa.

math.NT↗
arXiv2013

Newton polygons for a variant of the Kloosterman family

We study the p-adic valuations of roots of L-functions associated with certain families of exponential sums of Laurent polynomials in n variables over a finite field. The families we consider are reflection and Kloosterman variants of diagonal polynomials. Using decomposition theorems of Wan, we determine the Newton and Hodge polygons of a non-degenerate Laurent polynomial in one of these families.

math.NT↗
arXiv2012

Asymptotic variation of L-functions of exponential sums

Let Delta be an integral convex polytope containing the origin of dimension n in the n-dim real space and it is simplicial at all origin-less facets. Let A(Delta) be the space of all Laurent polynomials f parametered by its coefficients supported on the interior of Delta with with prescribed vertices. In this paper we prove that, if the interior points of Delta and its vertices generate almost all integral points in the cone of Delta, then there is a Zariski dense open subset U in A(Delta) defined over the rationals such that for every f in U(bar{Q}) and for p large enough the Newton polygon of the L function of exponential sum of f is equal to the generic Newton polygon for A(Delta)(bar{F_p}), and as p approaches infinity they both approach an absolute lower bound depending only on Delta. This paper also proves the following result for affine toric hypersurfaces: Let T_Delta be the space of all regular affine toric hypersurfaces given by f=0 for all Laurent polynomials f with given Newton polytope Delta. If the set of all integral points in Delta has a unimodular triangulation, then for p large enough the generic Newton polygon of T_Delta over bar{F_p} coincides with the Hodge polygon defined in terms of Hodge numbers of the toric family T_Delta determined solely by Delta.

math.NT↗
arXiv2010

The p-rank stratification of Artin-Schreier curves

We study a moduli space AS_g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification of AS_g by p-rank into strata AS_{g,s} of Artin-Schreier curves of genus g with p-rank exactly s. We enumerate the irreducible components of AS_{g,s} and find their dimensions. As an application, when p=2, we prove that every irreducible component of the moduli space of hyperelliptic k-curves with genus g and 2-rank s has dimension g-1+s. We also determine all pairs (p,g) for which AS_g is irreducible. Finally, we study deformations of Artin-Schreier curves with varying p-rank. Keywords: Artin-Schreier, hyperelliptic, curve, moduli, p-rank.

math.NT↗
arXiv2009

Crystalline representations of G_Qp^a with coefficients

This paper studies crystalline representations of G_K with coefficients of any dimension, where K is the unramified extension of Q_p of degree a. We prove a theorem of Fontaine-Laffaille type when σ-invariant Hodge-Tate weight less than p-1, which establishes the bijection between Galois stable lattices in crystalline representations and strongly divisible ϕ-lattice. In generalizing Breuil's work, we classify all reducible and irreducible crystalline representations of G_K of dimensional 2, then describe their mod p reductions. We generalize some results (of Deligne, Fontaine-Serre, and Edixhoven) to representations arising from Hilbert modular forms when σ-invariant Hodge-Tate weight less than p-1.

math.NT↗
arXiv2007

Hodge-Stickelberger polygons for L-functions of exponential sums of P(x^s)

Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of exponential sums of P(x^s) has a tight lower bound which we call `Hodge-Stickelberger polygon', depending only on d_1,d_2,s, and (p mod s). This Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for L-function of exponential sum of P(x) and the `Newton polygon' for L-function of exponential sum of x^s (given by the classical Stickelberger theory). We prove an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. We prove this Hodge-Stickelberger polygon is the limit of generic Newton polygons of P(x^s) in a sense that was made explicit in the paper.

math.NT↗
arXiv2006

Hyperelliptic curves over F_2 of every 2-rank without extra automorphisms

We prove that for any pair of integers 0\leq r\leq g such that g\geq 3 or r>0, there exists a (hyper)elliptic curve C over F_2 of genus g and 2-rank r whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties (A,λ) over F_2 of dimension g and 2-rank r such that \Aut(A,λ)={\pm 1}.

math.AG↗
arXiv2005

Asymptotic variation of L functions of one-variable exponential sums

Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number field generated by coefficients of f in Qbar. Let A^d be the dimension-d affine space over Qbar, identified with the space of coefficients of degree-d monic polynomials. Let NP(f mod P) denote the p-adic Newton polygon of L(f mod P;T). Let HP(A^d) denote the p-adic Hodge polygon of A^d. We prove that there is a Zariski dense open subset U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_{p-->oo} NP(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.

math.NT↗
arXiv2005

L-functions of Exponential sums over one-dimensional affinoid: Newton over Hodge

Let p be a prime and let F_pbar be the algebraic closure of the finite field of p elements. Let f(x) be any one variable rational function over F_pbar with n poles of orders d_1, ...,d_n. Suppose p is coprime to d_i for every i. We prove that there exists a Hodge polygon, depending only on d_i's, which is a lower bound to the Newton polygon of L functions of exponential sums of f(x). Moreover, we show that these two polygons coincide if p=1 mod d_i for every i=1,...,n. As a corollary, we obtain a tight lower bound of Newton polygon of Artin-Schreier curve.

math.NT↗