This article describes cubic function fields $L/K$ with prescribed ramification, where $K$ is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure $K'/K$ of $L/K$ is of genus zero, and a description of the twists of $L/K$ up to isomorphism over $K$. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on $K$. The article concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.
Let $K$ be a totally real field. In this article we present an asymptotic formula for the number of Hilbert modular cusp forms $f$ with given ramification at every place $v$ of $K$. When $v$ is an infinite place, this means specifying the weight of $f$ at $k$, and when $v$ is finite, this means specifying the restriction to inertia of the local Weil-Deligne representation attached to $f$ at $v$. Our formula shows that with essentially finitely many exceptions, the cusp forms of $K$ exhibit every possible sort of ramification behavior, thus generalizing a theorem of Khare and Prasad. From this …
The tame Gras-Munnier Theorem gives a criterion for the existence of a ${\mathbb Z}/{\mathbb Z}$-extension of a number field $K$ ramified at exactly a set $S$ of places of $K$ prime to $p$ (allowing real Archimedean places when $p=2$) in terms of the existence of a dependence relation on the Frobenius elements of these places in a certain governing extension. We give a new and simpler proof of this theorem that also relates the set of such extensions of $K$ to the set of these dependence relations. After presenting this proof, we then reprove the key Proposition 3 using the more sophisticated …
Let F be a totally real number field, n a prime integer, and G a unitary group of rank n defined over F that is compact at every infinite place. We prove an asymptotic formula for the number of cuspidal automorphic representations of G whose factors at finitely many places are prescribed up to inertia. The results and the methods used are a generalization to this setting of those used by Weinstein for GL_2.
We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases.
We compute the higher ramification groups and the Artin conductors of radical extensions of the rationals. As an application, we give formulas for their discriminant (using the conductor-discriminant formula). The interest in such number fields is due to the fact that they are among the simplest non-abelian extensions of the rationals (and so not classified by Class Field Theory). We show that this extensions have non integer jumps in the superior ramification groups, contrarily to the case of abelian extensions (as prescribed by Hasse-Arf theorem).
Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.
In this paper we study the moduli spaces $Simp^m_n$ of degree $n+1$ morphisms $ \mathbb{A}^1_{K} \to \mathbb{A}^1_{K}$ with "ramification length $<m$" over an algebraically closed field $K$. For each $m$, the moduli space $Simp^m_n$ is a Zariski open subset of the space of degree $n+1$ polynomials over $K$ up to $Aut (\mathbb{A}^1_{K})$. It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes -- here we are prescribing, instead, the ramification data. Exploiting the topological properties of the poset that encodes the ramification behaviour, we use a sheaf-theor…
In the present work, we determine explicitly the genus of any separable cubic extension of any global function field given the minimal polynomial of the extension. We give algorithms computing the ramification data and the genus of any separable cubic extension of any global rational function field.
We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend well-known properties of tame extensions to this more general setting; for instance, we show that these towers, when infinite, are ``asymptotically good'' (an explicit bound for the root discriminant is given). We study the difficult problem of bounding the relation-rank of the Galois groups in question. Results of Gordeev and Wingberg imply that the relation-rank can tend to infinity when the set of ramified primes is fixed but the leng…
We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a field. The splitting and ramification of places in a separable cubic extension of any global function field are completely determined, and precise Riemann-Hurwitz formulae are given. In doing so, we determine the decomposition of any cubic polynomial over a finite field.
We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow one to easily read ramification and splitting data from the generating equation, in analogy to the known theory for Artin-Schreier and Kummer extensions. We also describe explicit irreducibility criteria, integral bases, and Galois actions in terms of canonical generating equations.
We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $ξ$ such that $ξ+ξ^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(ξ)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.
We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension of $K$ where $K/\mathbb{Q}$ is cyclic of degree $3$. We also consider the group $G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the Cohen-Lenstra heuristics, we identify certain types of pro-$2$ group as the natural spaces where $G_2(K)$ and $G^+_2(K)$ live when the $2$-class group of $K$ is $2$-generated. While we do not have a theoretical scheme for assigning probabilities, we present data and make some observations and conjectures about the distribution of such groups.
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a place of E dividing v, we show that any field embedding E_S \to \bar{E_u} has a dense image. The "unramified outside S" number fields we use are cut out from the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main glob…
Let $K$ be a number field, which is tame and non totally real. In this article we give a numerical criterion, depending only on the ramification behavior of ramified primes in $K$, to decide whether or not the integral trace of $K$ is isometric to the integral trace of another number field $L$. As a byproduct of our proofs here, and in contrast with our previous results for cubic fields of positive discriminant, we show that for cubic fields of negative discriminant isometry between integral traces is equivalent to equality of discriminants.
In this note, we study linear determinantal representations of smooth plane cubics over finite fields. We give an explicit formula of linear determinantal representations corresponding to rational points. Using Schoof's formula, we count the number of projective equivalence classes of smooth plane cubics over a finite field admitting prescribed number of equivalence classes of linear determinantal representations. As an application, we determine isomorphism classes of smooth plane cubics over a finite field with 0, 1 or 2 equivalence classes of linear determinantal representations.
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.
We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Rédei's triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $\mathbb{Q}(\sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to Rédei's dihedral extension over $\mathbb{Q}$. We also give a cohomological interpretation of our symbols by…
Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a non-zero element $ξ\in \mathbb{F}_{q^{n}}$ such that $ξ+ ξ^{-1}$ is primitive and $T(ξ)=a, T(ξ^{-1})=b$, where $T(ξ)$ denotes the trace of $ξ$ in $\mathbb{F}_{q}$. This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of $n \geq 5$. A related stu…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is empty, or it contains curves of rank~0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of …
