arXiv subjects

Hui June Zhu

Publications and source records attributed to Hui June Zhu.

24 records · Page 2Linked to original sources

arXiv2005

Zeta functions of totally ramified p-covers of the projective line

In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary k poles of prescribed orders, such that for every geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of a p-cover of the projective line totally ramified at arbitrary k points of prescribed orders has an asymptotic generic lower bound.

math.NT↗

arXiv2003

Construction of some families of 2-dimensional crystalline representations

We construct explicitly some analytic families of etale (phi,Gamma)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular forms.

math.NT↗

arXiv2001

Families of supersingular curves in characteristic 2

This paper determines the normal forms of hyperelliptic supersingular curves of genus g over an algebraically closed field F of characteristic 2 for 0 < g< 9. We also show that every hyperelliptic supersingular curve of genus 9 over F has an equation y^2-y=x^19+c^8x^9+c^3x for some c in F_2bar. The paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus g < 10 over F_2bar.

math.AG↗

arXiv2001

Hyperelliptic curves in characteristic 2

In this paper we prove that there are no hyperelliptic supersingular curves over F_2bar of genus 2^n-1 for any integer n>1. Let g be a natural number, and h=floor(log_2(g+1)+1). Let X be a hyperelliptic curve over F_2bar of genus g>2 and 2-rank zero, given by an affine equation y^2-y=c_{2g+1} x^{2g+1} +...+ c_1 x. We prove that the first slope of the Newton polygon of X is bigger than or equal to 1/h. We also prove that the equality holds if (I) g<2^h-2, c_{2^h-1} is nonzero; or (II) g=2^h-2, c_{2^h-1} or c_{3(2^{h-1})-1} is nonzero. We prove that genus-4 hyperelliptic curve over F_2bar are precisely those with equations y^2 - y = x^9 + a x^5 + b x^3.

math.AG↗

arXiv2001

Slope estimates of Artin-Schreier curves

Let f(x) = x^d + a_{d-1}x^{d-1} + ... + a_0 be a polynomial of degree d in Q[x]. For every prime number p coprime to d and f(x) in (Z_p \cap Q)[x], let X/F_p be the Artin-Schreier curve defined by the affine equation y^p - y = f(x) mod p. Let NP_1(X/F_p) be the first slope of the Newton polygon of X/F_p. We prove that there is a Zariski dense subset U in the space A^d of degree-d monic polynomials over Q such that for all f(x) in U, the limit of NP_1(X/F_p) is equal to 1/d as p goes to infinity. This is a ``first slope version'' of a conjecture of Wan. Let X/F_pbar be an Artin-Schreier curve defined by the affine equation y^p - y = F(x) where F(x) = x^d + A_{d-1}x^{d-1} + ... + A_0. We prove that if p>d>1 then NP_1(X/F_pbar) >= ceiling((p-1)/d)/(p-1). If p>2d>3, we give a sufficient condition for the equality to hold.

math.AG↗

arXiv2000

On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field

We prove that for every field k and every positive integer n, there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k,n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n, the quantity S(F_q,n) approaches 1 as q approaches infinity over the prime powers.

math.AG↗