Determinant Expressions for Abelian Functions in Genus Two

Abelian points on algebraic curves

We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K^{ab} of K. We give: (i) an explicit family of diagonal plane cubic curves with Q^{ab}-points, (ii) for every number field K, a genus one curve C_{/Q} with no K^{ab}-points, and (iii) for every g \geq 4 an algebraic curve C_{/Q} of genus g with no Q^{ab}-points. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without (Q((t)))^{ab}-points.

Genus Field and Extended Genus Field of an Elementary Abelian Extension of Global Fields

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian $l$-extension of a field of rational functions, where $l$ is a prime number. In the Kummer case, if $K$ is contained in a cyclotomic funtion field, the construction is given using Leopoldt's ideas by means of Dirichlet characters. Following the definition of Anglès and Jaulent of extended Hilbert class field, we obtain the extended genus field of an elementary abelian $l$-extension of a field of rational functions.

18阶非阿贝尔群的整群行列式

The integer group determinants for the non-abelian groups of order 18

本研究针对18阶非阿贝尔群的整群行列式问题，完成了对该类群整群行列式的完整描述，填补了相关数论研究的空白。

Determinants of Subquotients of Galois Representations Associated to Abelian Varieties

Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell$, the $\ell^n$-torsion points of $A$ give rise to a representation $ρ_{A, \ell^n} : \gal(\bar{K} / K) \to \gl_{2g}(\zz/\ell^n\zz)$. In particular, we get a mod-$\ell$ representation $ρ_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\ff_\ell)$and an $\ell$-adic representation $ρ_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\zz_\ell)$. In this paper, we describe the possible determinants of subrepresentations (or more generally, subquotients) of these two representation for $\ell$ a prime number, as $A$ varies ove…

Genus fields of finite abelian extensions

In this paper we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the particular case of finite abelian $p$--extensions and give an explicit description of their genus field.

L-functions of Genus Two Abelian Coverings of Elliptic Curves over Finite Fields

Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields. Our goal is to provide a complete description of such a list.

Curves of every genus with many points, I: Abelian and toric families

Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we prove that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we use curves on toric surfaces to prove that liminf_{g-->infty}…

阿贝尔数域的亏格数分布

Distribution of genus numbers of abelian number fields

本文研究给定 Galois 群下数域阿贝尔扩张的亏格数分布，证明其平均值的渐近公式，并得出任一固定亏格数出现密度为零的结论。

有理函数域的阿贝尔扩张的广义亏格域

Extended genus fields of abelian extensions of rational function fields

本文研究了全局有理函数域有限阿贝尔扩张的广义亏格域，通过分析素数幂次循环扩张及分圆扩张复合域的性质，显式给出了基于“分圆投影”的一般有限阿贝尔扩张广义亏格域的求解结果。

来自亏格3双覆盖的小导子阿贝尔曲面

Abelian surfaces of small conductor from genus 3 double covers

通过亏格1曲线的亏格3双覆盖的Prym构造了约50万个Q上小导子阿贝尔曲面的数据库，并提供了对坏约化素数的控制方法。

18阶阿贝尔群的整数群行列式

The integer group determinants for the abelian groups of order 18

本文完整描述了18阶两个阿贝尔群的整数群行列式，包括循环行列式情形，从而完成了对小于20阶群的研究。

Computer search for curves with many points among abelian covers of genus 2 curves

Using class field theory one associates to each curve C over a finite field, and each subgroup G of its divisor class group, unramified abelian covers of C whose genus is determined by the index of G. By listing class groups of curves of small genus one may get examples of curves with many points; we do this for all curves of genus 2 over the fields of cardinality 5,7,9,11,13 and 16, giving new entries for the tables of curves with many points (www.manYPoints.org).

Galois group at Galois point for genus-one curve

We show all the possible structures of finite subgroups of the automorphism groups of elliptic curves. Here the automorphism means the biholomorphic transformation. Using the result, we determine every Galois group G at outer Galois point for genus-one curve with singular points. In particular, if G is abelian, then its order is at most nine. However if not so, the order is unbounded. Furthermore, we give every defining equation of the curve with the Galois point in the case where G is abelian.

Integer group determinants for abelian groups of order 16

For any positive integer $n$, let ${\rm C}_{n}$ be the cyclic group of order $n$. We determine all possible values of the integer group determinant of ${\rm C}_{4} \times {\rm C}_{2}^{2}$, which is the only unsolved abelian group of order $16$.

Split abelian surfaces over finite fields and reductions of genus-2 curves

For prime powers q, let s(q) denote the probability that a randomly-chosen principally-polarized abelian surface over the finite field F_q is not simple. We show that there are positive constants B and C such that for all q, B (log q)^{-3}(log log q)^{-4} < s(q)sqrt(q) < C (log q)^4(log log q)^2, and we obtain better estimates under the assumption of the generalized Riemann hypothesis. If A is a principally-polarized abelian surface over a number field K, let pi_split(A/K, z) denote the number of prime ideals p of K of norm at most z such that A has good reduction at p and A_p is not simple. W…

Abelian surfaces of GL2-type as Jacobians of curves

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian varieties are isomorphic to Weil's restriction of quadratic \Q-curves, and present examples.

Function field genus theory for non-Kummer extensions

In this paper we first obtain the genus field of a finite abelian non-Kummer $l$--extension of a global rational function field. Then, using that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields and our previous results, we deduce the general expression of the genus field of a finite abelian extension of a global rational function field.

Ruelle type L-functions versus determinants of Laplacians for torsion free abelian groups

We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus $\bR^{\n}/\G$ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.

Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves

Let k be a field of characteristic different from 2. There can be an obstruction for an indecomposable principally polarized abelian threefold (A,a) over k to be a Jacobian over k. It can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numeric results to prove or refute the existence of some optimal curves of genus 3.

Class fields, Dirichlet characters and extended genus fields of global function fields

We obtain the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses class field theory. First we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields and compare this approach with the one given by …

Genus fields of abelian extensions of congruence rational function fields

In the published version of this paper [Finite Fields and Their Applications {\bf 20} (2013) 40--54], there is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statments for Theorems 4.2, 4.5 and 5.2 We give a construction of genus fields for congruence function fields. First we consider the cyclotomic function field case following the ideas of Leopoldt and then the general case. As applications we give explicitly the genus fields of Kummer, Artin--Schreier and cyclic $p$--extensions. Kummer extensions were obtained previously by G. Peng and Artin…

p^k-torsion of genus two curves over F_{p^m}

We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves.

Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties

We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations. For genus 2 we give a formula for the full Eisenstein cohomology.

Sato-Tate groups of genus 2 curves

We describe the analogue of the Sato-Tate conjecture for an abelian variety over a number field; this predicts that the zeta functions of the reductions over various finite fields, when properly normalized, have a limiting distribution predicted by a certain group-theoretic construction related to Hodge theory, Galois images, and endomorphisms. After making precise the definition of the "Sato-Tate group" appearing in this conjecture, we describe the classification of Sato-Tate groups of abelian surfaces due to Fite-Kedlaya-Rotger-Sutherland. (These are notes from a three-lecture series present…

25阶初等阿贝尔群的5-可整除整数群行列式

The $5$-divisible integer group determinants for the elementary abelian group of order 25

本文针对25阶初等阿贝尔群C₅×C₅，证明其5-可整除群行列式的反向包含关系，结合已有结果完成25阶群的Taussky-Todd整数群行列式问题分类。

Continued fractions and Hankel determinants from hyperelliptic curves

Following van der Poorten, we consider a family of nonlinear maps which are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus $\mathrm{g}$. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the simplest case $\mathrm{g}=1$ this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in a particular form by Chang, Hu and Xin, We extend these formulae to the higher genus case, and prove that generic Hankel determinants in…

Endomorphism algebras of geometrically split genus 2 Jacobians over Q

The main result of [FG20] classifies the 92 geometric endomorphism algebras of geometrically split abelian surfaces defined over Q. We show that 54 of them arise as geometric endomorphism algebras of Jacobians of genus 2 curves defined over Q, and that the remaining 38 do not. In particular, we exhibit 38 examples of Qbar-isogeny classes of abelian surfaces defined over Q that do not contain any Jacobian of a genus 2 curve defined over Q, and for each of the 54 algebras that do arise we exhibit a curve realizing them.

The Integer group determinants for the Heisenberg group of order $p^3$

We establish a congruence satisfied by the integer group determinants for the non-abelian Heisenberg group of order $p^3$. We characterize all determinant values coprime to $p$, give sharp divisibility conditions for multiples of $p$, and determine all values when $p=3$. We also provide new sharp conditions on the power of $p$ dividing the group determinants for $\mathbb Z_p^2$. For a finite group, the integer group determinants can be understood as corresponding to Lind's generalization of the Mahler measure. We speculate on the Lind-Mahler measure for the discrete Heisenberg group and for tw…

Generating a Quadratic Forms from a Given Genus

Given a non-empty genus in $n$ dimensions with determinant $d$, we give a randomized algorithm that outputs a quadratic form from this genus. The time complexity of the algorithm is poly$(n,\log d)$; assuming Generalized Riemann Hypothesis (GRH).

24阶阿贝尔群的整数群行列式

Integer group determinants for abelian groups of order 24

本文通过特征分解和有限检验，确定了三个24阶阿贝尔群的整数群行列式集合，并证明了奇行列式集合的包含关系及无限族差异。

Genus 3 hyperelliptic curves with CM via Shimura Reciprocity

Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton d…

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and $n$-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szegö kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identi…

Abelian Surfaces over totally real fields are Potentially Modular

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

亏格2曲线雅可比簇的Edwards模型类比

An analog of the Edwards model for Jacobians of genus 2 curves

本文给出了亏格2曲线雅可比簇的P^3 x P^3嵌入显式方程，作为椭圆曲线Edwards模型的自然类比，比标准P^15版本更简洁，并在特定条件下具有普遍群定律。

二次域上具有处处好约化的亏格2曲线

Genus two curves with everywhere good reduction over quadratic fields

本研究针对二次域上二维绝对单阿贝尔簇的存在性问题，构造了首个由实/复二次域与对应域上处处好约化亏格2曲线组成的无穷序列，并证明其雅可比簇为绝对单阿贝尔簇。

Rationality of twists of the Siegel modular variety of genus $2$ and level $3$

Let $\overlineρ: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the $3$-torsion of a principally polarized abelian surface $A/\mathbf{Q}$. We prove that the moduli space $\mathcal{A}_2(\overlineρ)$ of principally polarized abelian surfaces $B/\mathbf{Q}$ admitting a symplectic isomorphism $B[3] \simeq \overlineρ$ of Galois representations is never rational over $\mathbf{Q}$ when $\overlineρ$ is surjective, even though it is both ratio…

计算某些零维 Shimura 曲线上的点数

Counting points on some genus zero Shimura curves

本文利用零维 Shimura 曲线上有界高度点的计数，研究数域 $K$ 上具潜在四元数乘法的阿贝尔曲面数量。

Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato--Tate groups over a number field

Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\geq 1$ that is isogenous over $\overline{\mathbb{Q}}$ to $E^g$, where $E$ is an elliptic curve. If $E$ does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic $\mathbb{Q}$-curves $E$ is isogenous to a curve defined over a polyquadratic extension of $\mathbb{Q}$. We show that one can adapt Ribet's methods to study the field of definition of $E$ up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato--Tate groups: First, we show that $1…

p-adic 伪测度的亏格理论——Tame 核与阿贝尔 p-分歧

Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification

本文利用 p-adic 伪测度的亏格理论，比较实阿贝尔域中 tame 核与阿贝尔 p-分歧挠群，证明 Deng--Li 猜想并给出 p-秩下界。