Douglas Ulmer
Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
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We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.
We give a necessary and sufficient condition in terms of a matrix for which all Tate classes are Lefschetz for simple abelian varieties over an algebraic closure of a finite field. As an application, we prove under an assumption that all Tate classes are Lefschetz for simple factors of Fermat Jacobian of prime degree.
We study the Abel-Jacobi image of the Ceresa cycle W_k-W_k^-, where W_k is the image of the k-th symmetric product of a curve X on its Jacobian variety. For the Fermat curve of degree N, we express it in terms of special values of generalized hypergeometric functions and give a criterion for the non-vanishing of W_k-W_k^- modulo algebraic equivalence, which is verified numerically for some N and k.
Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. We compare three classifications of $BT_1$ group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl--Oort types of Fermat quotient curves and we compute four invariants of the $p$-torsion group schemes of these curves.
In this paper we construct certain higher Chow cycles in the $K_{1}$ of the Jacobian of Fermat curves, generalising a construction of Collino. We further compute the regulator of these elements in terms of special values of hypergeometric functions. Otsubo has computed the regulator of certain elements of $K_0$ and $K_2$ of Fermat varieties and this paper is along the same lines.
The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat curve. We give examples of torsion points whose associated Galois structure is trivial, as well as points of infinite order whose associated Galois structure is non-trivial.
Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=δ$ where $δ$ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When $p\nmid δ$, we bound the Selmer groups of these Jacobian varieties. Moreover, if $p$ is regular and all prime ideals of $K$ dividing $δ$ are inert in $K/F$, the Selmer groups are explicitly determined an…
We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a constant multiple of its dimension. In the case of the tower of Fermat curves, we show that the constant can be taken arbitrarily close to 1. The main result is used in the forthcoming paper of Guillermo Mantilla-Soler on the Mordell-Weil rank of the modular Jacobian J(Np^m).
Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians due to Zarhin to give many examples of higher dimensional, absolutely simple Jacobians over $k(t)$ with bounded rank in towers $k(t^{1/p^r})$. In many cases we are able to compute the rank at every layer of the tower.
Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(τ,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/τB(k)$ varies when we take an unramified abelian cover $π:C'\to C$ defined over $k$.
We use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8-th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic…
Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
本文计算了超椭圆曲线y^2=x^m+1的Jacobian簇J_m的等元分解、端元定义域和连通单值域，并展示了高维退化现象。
An Exploration of Degeneracy in Abelian Varieties of Fermat Type
本文研究费马类型阿贝尔簇的退化现象，通过Mumford-Tate群、Hodge群和Sato-Tate群分析退化特征，证明当m为奇合数时，超椭圆曲线的雅可比簇退化。
Torsion cycles on Fermat varieties
本文通过埃尔利克的方法重新证明了罗赫利奇关于费马曲线的定理，并将其推广到更高维度的无同源循环和高智周期。
The five-color hypercube Adinkra and the Jacobian of a generalized Fermat curve
本文研究了五色超立方体Adinkra与广义费马曲线雅可比之间的映射，提出了一种组合算法计算高度函数图像，并给出了生成分歧系数的严格界限。
We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the $L$-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients.
We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the Mordell--Weil group of the Jacobian variety explicilty. We show that the degree $0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $3$. Thus the relative Brauer group is non-trivial. We also show that this quartic violates the local-global property for linear determinantal representations.
Given a prime $\ell\geq 3$ and a positive integer $k \le \ell-2$, one can define a matrix $D_{k,\ell}$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian ${\mathrm Jac}({\mathcal C}_{k,\ell})$ of a certain quotient of the Fermat curve of exponent $\ell$. For a fixed $\ell$, the existence of $k$ for which $D_{k,\ell}$ is singular (equivalently, for which the rank of the Hodge group of ${\mathrm Jac}({\mathcal C}_{k,\ell})$ is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such…
In this paper we derive a recursion for the zeta function of each function field in the second Garcia-Stichtenoth tower when $q=2$. We obtain our recursion by applying a theorem of Kani and Rosen that gives information about the decomposition of the Jacobians. This enables us to compute the zeta functions explicitly of the first six function fields.
Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. Our main result is that every $BT_1$ scheme group over $k$ occurs as a direct factor of the $p$-torsion group scheme of the Jacobian of an explicit curve defined over $\mathbb{F}_p$. We also treat a variant with polarizations. Our main tools are the Kraft classification of $BT_1$ group schemes, a theorem of Oda, and a combinatorial description of the de Rham …
Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute th…
We study the endomorphism ring $End(J(C))$ of the complex jacobian $J(C)$ of a curve $y^p=f(x)$ where $p$ is an odd prime and $f(x)$ is a polynomial with complex coefficiens of degree $n>4$ and without multiple roots. Assume that all the coefficients of $f$ lie in a (sub)field $K$ and the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$. Then we prove that $End(J(C))$ is the ring of integers in the in the $p$th cyclotomic field, if $p$ is a Fermat prime (e.g., $p=3,5,17,257$). Similar results for $p=2$ (the case of hyperelliptic curves) were …
Jacobians of Graphs via Edges and Iwasawa Theory
本研究类比数域类群与图的雅可比不变量，针对连通图的$\mathbb{Z}_p$塔构造与雅可比相关的岩泽模，并给出用于计算该塔中雅可比渐近规模的实例。
We study a natural question in the Iwasawa theory of algebraic curves of genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field $K$ of genus $g>1$, such that the Jacobian of $X$ has good ordinary reduction at the primes above $p$. Fix an odd prime $p$ and for any integer $n>1$, let $K_n^{(p)}$ denote the degree-$p^n$ extension of $K$ contained in $K(μ_{p^{\infty}})$. We prove explicit results for the growth of $\#X(K_n^{(p)})$ as $n\rightarrow \infty$. When the Jacobian of $X$ has rank zero and the associated adelic Galo…
We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is geometrically isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato-Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients.
Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the $p$-adic loop group and the formal group.
The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected graph $X$; it is a finite abelian group whose cardinality is equal to the number of spanning trees of $X$ (Kirchhoff's Matrix Tree Theorem). A specific type of covering graph, called a derived graph, that is constructed from a voltage graph with voltage group $G$ is the object of interest in this paper. Towers of derived graphs are studied by using aspects of classical Iwasawa Theory (from number theory). Formulas for the orders of the Sylow $p$-subgroups of Jacobians in an …
The topic of this paper concerns a certain relation between the jacobians of various quotients of the modular curve $X(p)$, which relates the jacobian of the quotient of $X(p)$ by the normaliser of a non-split Cartan subgroup of $GL_2(F_p)$ to the jacobians of more standard modular curves. In this paper, we confirm a conjecture of Merel found in a paper of Darmon-Merel, "Winding quotients and some variants of Fermat's Last Theorem", Crelle, v. 490, p. 81-100, 1997, which describes this relation in terms of explicit correspondences. The method used is to reduce the conjecture to showing a certa…
We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the affine part of Jacobian varieties, we give an algorithm to obtain all linear determinantal representations up to equivalence. We also report our recent study on computations of linear determinantal representations of twisted Fermat cubics defined over the field of rational numbers. This paper is a summary of the author's talk at the JSIAM JANT workshop on algorit…
Let $k$ be a field of characteristic $0$, and let $α_{1}$, $α_{2}$, ..., $α_{5}$ be algebraically independent and transcendental over $k$. Let $K$ be the transcendental extension of $k$ obtained by adjoining the elementary symmetric functions of the $α_{i}$'s. Let $J$ be the Jacobian of the hyperelliptic curve defined over $K$ which is given by the equation $y^{2} = \prod_{i = 1}^{5} (x - α_{i})$. We define a tower of field extensions $K = K_{0}' \subset K_{1}' \subset K_{2}' \subset ...$ by giving recursive formulas for the generators of each $K_{n}'$ over $K_{n - 1}'$, and let $K_{\infty}' =…
We investigate a novel geometric Iwasawa theory for $\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over $k$ associated to a $\mathbf{Z}_p$-extension of function fields totally ramified over a finite non-empty set of places, we investigate the growth of the $p$-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty$. By Dieudonné theory, this amounts to stud…
A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We prove that for each integer g at least 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of the new part …
Unramified Geometric Iwasawa theory of function fields
本文研究特征$p>0$代数闭域上曲线非分歧$\mathbf{Z}_p$-塔的雅可比簇Newton多边形，证明其新部分斜率最终为$p^{-n}$的有理系数仿射函数。
We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil…
Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields
本文证明某些 Hecke 特征在分圆域上的非零性，通过计算特征函数和 Weil 表示作用，应用于 Fermat 曲线和超椭圆曲线的 Jacobian，得到无穷多个解析秩为零的 twist。
Iwasawa theory and the representations of finite groups
本文通过表示论精细化有限凯莱图的岩泽理论，基于图zeta函数与数论L函数的类比研究凯莱图ℤ_ℓ-塔及雅可比行列式渐近增长，证明岩泽多项式可通过基础群不可约表示典范分解并定义表示论岩泽多项式。
Let $k$ be a perfect field of characteristic $p$ and $Γ$ an infinite, first countable pro-$p$ group. We study the behavior of the $p$-primary part of the "motivic class group", i.e. the full $p$-divisible group of the Jacobian, in any $Γ$-tower of function fields over $k$ that is unramified outside a finite (possibly empty) set of places $Σ$, and totally ramified at every place of $Σ$. When $Σ=\emptyset$ and $Γ$ is a torsion free $p$-adic Lie group, we obtain asymptotic formulae which show that the $p$-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $Σ\…
Let $C_k$ be a smooth projective curve over a global field $k$, which is neither rational nor elliptic. Harris-Silverman, when $p=0$, and Schweizer, when $p>0$ together with an extra condition on the Jacobian variety $\operatorname{Jac}(C_k)$ arising from Mordell's conjecture, showed that $C$ has infinitely many quadratic points over some finite field extension $L/k$ inside $\overline{k}$ (a fixed algebraic closure of $k$) if and only if $C$ is hyperelliptic or bielliptic. Now, let $C_k$ be a smooth plane curve of a fixed degree $d\geq4$ with $p=0$ or $p>(d-1)(d-2)+1$ (up to an extra condition…
One of the most significant discrete invariants of a quadratic form $ϕ$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behaviour of $ϕ$ under scalar extension to arbitrary overfields of $k$. A similarly important, but more accessible variant of this notion is that of the Knebusch splitting pattern of $ϕ$, which captures the isotropy behaviour of $ϕ$ as one passes over a certain prescribed tower of $k$-overfields. In this paper, we determine all possible values of this latter invariant in the case where $ϕ$ is totally singul…
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