Niko Naumann
In [P] R. Pellikaan introduced a two variable zeta-function for a curve over a finite field and proved that it is a rational function. Here we show that its denominator is absolutely irreducible. This is motivated by work of J. Lagarias and E. Rains on an analogous two variable zeta-funtion for number fields.
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We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the ``generic'' irreducibility of the numerator of the zeta functions in a family of curves with large monodromy.
Irreducibility criteria for the preimages of a transverse variety under endomorphisms of products of elliptic curves
本文针对椭圆曲线乘积中横截子簇在对角自同态下的原像，给出了两种不可约性证明（算术与几何），并应用于多项式族不可约性、高度估计及本质极小值下界。
We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.
Let C be an arithmetically Cohen-Macaulay curve of arithmetic genus 4. We prove that the family of such curves of degree 8 in $P^4_k$ is irreducible.
The purpose of this paper is to give an algorithm for deciding the irreducibility of reduced algebroid curves over an algebraically closed field of arbitrary characteristic. To do this, we introduce a new notion of local tropical variety which is a straightforward extension of tropism introduced by Maurer, and then give irreducibility criterion for algebroid curves in terms local tropical varieties. We also give an algorithm for computing the value-semigroups of irreducible algebroid curves. Combining the irreducibility criterion and the algorithm for computing the value-semigroups, we obtain …
We show that for any numerical semigroup H of genus g at most 6, the locus of Weierstrass points on curves of genus g with Weierstrass semigroup H is irreducible and that for all but possibly two semigroups it is stably rational.
The irreducibility of the spaces of rational curves on del Pezzo manifolds
本文通过研究赋值映射纤维，证明 Picard 秩 1、维数至少 4 的 del Pezzo 流形上有理曲线空间不可约，并由此推出几何 Manin 猜想。
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\PP^r$. In this note, we show that any non-empty $\mathcal{H}_{g+2,g,4}$ is irreducible without any restriction on the genus $g$. Our result augments the irreducibility result obtained earlier by Hristo Iliev(2006), in which several low genus $g\le 10$ cases have been left untreated.
Let $C$ be a complex projective smooth curve and $W$ a symplectic vector bundle of rank $2n$ over $C$. The Lagrangian Quot scheme $LQ_{-e}(W)$ parameterizes subsheaves of rank $n$ and degree $-e$ which are isotropic with respect to the symplectic form. We prove that $LQ_{-e}(W)$ is irreducible and generically smooth of the expected dimension for all large $e$, and that a generic element is saturated and stable.
In this short note, I point out that results of Ballico and Kool--Shende--Thomas together imply that on $K3$, Enriques, and Abelian surfaces, if $L$ is a very ample and $(2p_a(L)-2g-1)$-spanned line bundle, then the equigeneric Severi variety $V_{g}(L)$ of all curves in $|L|$ having genus $g$ is non-empty, irreducible, of the expected dimension, and its general member is a $(p_a(L)-g)$-nodal curve.
We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a maximal curve over a finite field.
Let $X$ be a smooth projective surface and $L\in \mathrm{Pic}(X)$. We prove that if $L$ is $(2k-1)$-spanned, then the set $\tilde{V}_k(L)$ of all nodal and irreducible $D\in |L|$ with exactly $k$ nodes is irreducible. The set $\tilde{V}_k(L)$ is an open subset of a Severi variety of $|L|$, the full Severi variety parametrizing all integral $D\in |L|$ with geometric genus $g(L)-k$.
In math.AG/0108089 we gave sufficient conditions for the irreducibility of the family V of irreducible curves in the linear system |D| with precisely r singular points of topological respectively analytical types S1,...,Sr on several classes of smooth projective surfaces S. The conditions where of the form (tau'(S1)+2)^2 +...+ (tau'(Sr)+2)^2 < c*(D-K)^2, where tau' is some invariant of singularity types, K is the canonical divisor of S and c is some constant. In the present paper we improve this condition, that is the constant c, by a factor 9.
We determine what isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and to count the number of curves, up to k-isomorphism, leading to the same zeta function.
In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $E_1\times E_2$ without complex multiplication are integers. By studying elliptic curves on $E_1\times E_2$ we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $E_1\times E_2$ are integers if and only if the minimal degree of an isogeny $E_1\to E_2$ equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $E_1\times E_2$.
The numerical data of an embedded resolution determine the candidate poles of Igusa's p-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case.
Riemann Hypothesis for Non-Abelian Zeta Functions of Genus 2 Curves
本文针对有限域上亏格2曲线的Weng非阿贝尔zeta函数，利用半稳定丛模空间几何性质、高秩结果与渐近分析，证明了当秩n趋于无穷时黎曼猜想成立。
We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each connected component. In the course of the proof we study the class of a Severi-Brauer scheme over a general base in the Grothendieck ring of varieties.
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves
本研究将有理数域上椭圆曲线的Sato-Tate猜想与murmuration现象从阿贝尔Hasse-Weil Zeta函数推广至其p约化的非阿贝尔高秩Zeta函数，证明二者在该场景下同样成立。
Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring O_{P,X} at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if O_{P,X} is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincare series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the gene…
This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.
We prove the irreducibility of the moduli space of rank 2 semistable torsion free sheaves (with a generic polarization and any value of c_2) on a K3 or a del Pezzo surface. In the case of a K3 surface, we need to prove a result on the connectivity of the Brill-Noether locus for singular curves on the surface. In the case of a del Pezzo surface, we reduce the problem to the case of P^2 by first relating the moduli spaces of the plane and the blown-up plane, and then studying how the moduli space changes when we change the polarization.
We prove irreducibility for the space of cyclic covers of fixed numerical type between smooth projective curves, and also for the space of cyclic covers of prime order and of fixed numerical-combinatorial type between moduli-stable projective curves. As an application we describe the irreducible components of the singular locus of the compactified moduli space of curves $\bar{\mahfrak M_g}$, extending the work of Cornalba, who described the irreducible components of the singular locus of the moduli space of curves $\mahfrak M_g}$.
Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb P^r$. A component of $\mathcal{H}_{d,g,r}$ is rigid in moduli if its image under the natural map $π:\mathcal{H}_{d,g,r} \dashrightarrow \mathcal{M}_{g}$ is a one point set. In this note, we provide a proof of the fact that $\mathcal{H}_{d,g,r}$ has no components rigid in moduli for $g > 0$ and $r=3$, from which it follows that the only smooth projective curves embedded in $\m…
The irreducibility of the moduli space of pointed stable curves with dormant $\mathrm{PGL}_2^{(N)}$-oper
本文研究素特征下带 dormant $\text{PGL}_2^{(N)}$-oper 的带点稳定曲线模空间，通过引入基于 $p^N$-曲率特征多项式的高阶 Hitchin-Mochizuki 态射，证明了该模空间的不可约性。
In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space compl…
Zeta-functions of Curves over Finite Fields
本文研究有限域上给定亏格曲线的 zeta 函数构造，利用椭圆曲线 Jacobian 乘积与目标曲线 Jacobian 的同源条件及 Frobenius 特征多项式确定其 zeta 函数与有理点数。
The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces
本文通过热带几何方法，证明了在任意代数闭域上经典Hurwitz空间的不可约性，并建立了丰富托里曲面类（包括所有经典托里曲面）上Severi簇在任意特征下的不可约性。
In this paper we give a criterion of irreducibility for a complex power series in two variables, using the notion of jacobian Newton diagrams, defined with respect to any direction. Moreover we study the singularity at infinity of a plane affine curve with one point at infinity for which the global counterpart of our main result holds.
We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold fo…
Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields
该研究针对有限域上超椭圆曲线的函数域，证明了Cohen-Lenstra型类群分布结果与ζ函数消失的稀疏性结果，通过推广的Hurwitz概形点计数技术及酉大单值群定理完成证明。
In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch.
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis
本文提出不依赖黎曼假设的新方法，利用更强渐近界建立椭圆曲线高阶Zeta函数的Sato-Tate定律并表述Murmuration现象。
Irreducibility of Some Quot Schemes on Nodal Curves
本文证明了对于算术亏格 $g \geqslant 2$ 的整射影节点曲线 $C$，当次数 $d$ 足够大时，向量丛 $E$ 的 Quot scheme $\textrm{Quot}_{C/\mathbb C}(E,k,d)$ 是不可约的。
We define a divisorial motivic zeta function for stable curves with marked points which agrees with Kapranov's motivic zeta function when the curve is smooth and unmarked. We show that this zeta function is rational, and give a formula in terms of the dual graph of the curve.
In 1985 Joe Harris proved the long standing claim of Severi that equisingular families of nodal plane curves are irreducible whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor D on a smooth projective surface S it thus makes sense to look for conditions which ensure that the family V_D(S_1,...,S_r) of irreducible curves in the linear system |D| with precisely r singular points of types S_1,...,S_r is irreducible. Considering different surfaces including general surfaces in P^3 and products of curves, we produce a sufficient con…
In this paper, we introduce (local and) global non-abelian zeta functions for general curves. As an example, we compute the so-called rank two zeta functions for genus two curves by studying non-abelian Brill-Noether loci and their infinitesimal structures.
Uniform irreducibility of Galois action on the $\ell$-primary part of Abelian $3$-folds of Picard type
本文研究了皮卡型阿贝尔3折的ℓ-幂部分上格罗森德克尔作用的均匀不可约性，并将其推广到具有虚二次域乘法的二维阿贝尔3折族。
We give necessary and sufficient conditions for moduli spaces of semistable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the nilpotent cone of GL_n-Higgs bundles and the irreducible components of moduli of systems of Hodge bundles on curves. As we do not impose coprimality restrictions, we can apply this to prove connectedness for moduli spaces of U(p,q)-Higgs bundles.
We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.
The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other.
To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups. In lower ranks, we show that these two types of zetas coincide and satisfy the Riemann Hypothesis. For general cases, exposed is an intrinsic relation on automorphism groups of semi-stable bundles over elliptic curves, the so-called counting miracle. All this, together with Harder-Narasimhan, Desale-Ramanan and Zagier's result, gives an effective way to coun…
New local and global non-abelian zeta functions for elliptic curves are studied using certain refined Brill-Noether loci in moduli spaces of semi-stable bundles. Examples of these zeta functions and a justification of using only semi-stable bundles are given too. We end this paper with an appendix on the so-called Weierstrass Groups for general curves, which is motivated by a construction of Euler systems from torsion points (of elliptic curves).
Using the sieve for Frobenius, we show that, in a certain sense, the roots of the L-functions of "most" algebraic curves over finite fields do not satisfy any non-trivial (linear or multiplicative) rational dependency relations. This can be seen as an analogue of conjectures of linear independence among ordinates of zeros of L-functions over number fields. As a corollary, we find, for "most" pairs of distinct algebraic curves over a finite field, the limiting distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The meth…
The Poincare series of an irreducible plane curve singularity equals the zeta function of its monodromy, by a result of Campillo, Delgado and Gusein-Zade. We derive this fact from a formula of Ebeling and Gusein-Zade relating the Poincare series of a quasi-homogeneous complete intersection singularity to the Saito dual of a product of zeta functions.
In general, the L-polynomial of a curve of genus $g$ is determined by $g$ coefficients. We show that the L-polynomial of a supersingular curve of genus $g$ is determined by fewer than $g$ coefficients.
It is very fundamental to study irreducible plane curve singularities in algebraic geometry. The contents of the paper consist of three parts, called Part[A], Part[B] and Part[C] with Good Appendix. Our aim is to prove by Part[B] and Part[C] that a complete irreducibility algorithm for the Weierstrass polynomial of two complex variables and the Puiseux expansions in Part[A] can be explicitly and rigorously computable in an elementary way, as follows. For brevity, Weierstrass polynomials may be written by W-polys throughout this paper.
In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence explains in a certain sense) our work on non-abelian zeta functions for number fields.
We study the irreducible components of special loci of curves whose group of symmetries is given as certain group extension. We introduce some relative Hurwitz data, which we show by using mixed étale cohomology theory, identifies some irreducible components for rational and normal non-abelian special loci and Hurwitz spaces. A heuristic, that is supported by three classes of examples, provides an additional context for building further irreducible loci.
Motivic Coh and Quot zeta functions of singular curves
本文建立代数框架研究奇异曲线上Quot与Coh zeta函数，证明其motivic有理性和反射原理，并揭示与Rogers--Ramanujan型q-级数的新联系。
The auto-Igusa zeta function of a plane curve singularity is rational
研究证明平面曲线奇点芽的自 Igusa 函数具有有理性，并由此给出特征零代数闭域上平面曲线在一点光滑的新判据。
In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to th…
The motivic Hilbert zeta function of a variety is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points of the variety. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.
Hilbert irreducibility for algebraic points
研究数域上曲线覆盖中，给定次数d的代数点使得纤维可约的集合有限性条件，并证明在分支点足够多时该集合有限。
For each $(m+1)$-tuple ${\bf n}_m=(n_0,n_1,\ldots,n_m)$ of positive integers, the ${\bf n}_m$-derived zeta function $\widehatζ_{X,\mathbb F_q}^{\,({\bf n}_m)}(s)$ is defined for a curve $X$ over $\mathbb F_q$. This derived zeta function satisfies standard zeta properties. In particular, similar to the Artin Zeta function of $X/\mathbb F_q$, this ${\bf n}_m$-derived Zeta function of $X$ over $\mathbb F_q$ is a ratio of a degree $2g$ polynomial $P_{X,\mathbb F_q}^{({\bf n}_m)}$ in $T_{{\bf n}_m}=q^{-s\prod_{k=0}^mn_k}$ by $(1-T_{{\bf n}_m})(1-q_{{\bf n}_m}T_{{\bf n}_m})T_{{\bf n}_m}^{g-1}$ with …
The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For nondegenerate functions it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, Veys showed how to filter the actual poles out of the candidate poles induced by the resolution graph. In this note we show how to determine from the Newton polyhedron of a nondegenerat…
The $p$-adic zeta function of a plane curve singularity
本研究借助环面修改与相容性，在解析变量替换公式下计算平面曲线奇点的局部 $p$-进 zeta 函数，规避了代数情形与 Denef 公式。
In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture by Weng for the ranks 2 and 3, and we prove the countin…
We give a natural construction of unramified over Z framed mixed Tate motives, whose periods are the multiple zeta values. Namely, for each convergent multiple zeta-value we define two boundary divisors A and B in the moduli space M_{0,n+3} of stable curves of genus zero. The corresponding multiple zeta-motive is the n-th cohomology of the pair (M_{0,n+3} -A,B).
We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than deg(F). The algorithm uses the theory of approximate roots and may be seen as a generalization of Abhyankhar's irreducibility criterion to the case of non algebraically closed residue fields. More generally, we show that we can test within the same complexity if a polynomial is pseudo-irreducible, a larger class of polynomials containing irreducible one…
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