In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with "ordinary good" or "ordinary semistable" reduction at a given prime. We also give new proofs of: (1) The Manin-Mumford conjecture: There are only finitely many torsion points lying on a curve of genus at least 2 embedded in its Jacobian by an Albanese map; and (2) The Coleman-Kaskel-Ribet conjecture: If p is a prime number which is at least 23, then the only torsion points lying on the curve X_0(p), embedded in its Jacobian by a cuspidal embedding, are the cusps (together with the hyperelliptic branch points when X_0(p) is hyperelliptic and p is not 37). In an effort to make the exposition as useful as possible, we provide references for all of the facts about modular curves which are needed for our discussion.
We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion of an `almost rational torsion point.'
We prove that the groups PSL(r,q^d) can be realized F_q(T)-regularly as Galois groups over the purely transcendental field F_q(T)(t_1,...,t_{r-1}) if r is even and r/2 is coprime to q^d-1. The method is to use twisted moduli schemes of Drinfeld modules (a la Shih). If one has sufficient control over modular forms, it can be made into an algorithm to find explicit equations. We do this for PSL(2,q^2) using Drinfeld modular forms.
We prove that, on average, elliptic curves over Q have finitely many primes p for which they possess a p-adic point of order p. We include a discussion of applications to companion forms and the deformation theory of Galois representations.
We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.
Let $ρ$ be a mod $\ell$ Galois representation. We show how to compute $ρ$, given the characteristic polynomial of the image of the Frobenius at one prime $p$ and a curve $C$ whose Jacobian contains $ρ$ in its $\ell$-torsion. The main ingredient is a method to $p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.
Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρ_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of ρ_E for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly disjoint from the cyclotomic extension of Q, then ρ_E will be surjective for "most" elliptic curves over k.
We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some "generalized Hasse invariants" adapted to the Ekedahl-Oort stratification of the special fiber.
We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.
Let $F$ be a totally real number field and let $p$ be a prime unramified in $F$. We prove the existence of Galois pseudo-representations attached to mod $p^m$ Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for $F$ of level prime to $p$.
To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explici…
Fix a positive integer $g$ and rational prime $p$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $p$ representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of $\mathbb{Q}$.
Let $F$ be a local non-Archimedean field with ring of integers $o$. Let $\bf X$ be a one-dimensional formal $o$-module of $F$-height $n$ over the algebraic closure of the residue field of $o$. By the work of Drinfeld, the universal deformation $X$ of $\bf X$ is a formal group over a power series ring $R_0$ in $n-1$ variables over the completion of the maximal unramified extension of $o$. For $h \in \{0,...,n-1\}$ let $U_h$ be the subscheme of $\Spec(R_0)$ where the connected part of the associated divisible module of $X$ has height $h$. Using the theory of Drinfeld level structures we show tha…
We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic is…
Given a rational variety $V$ defined over $K$, we consider a principally polarized abelian variety $A$ of dimension $g$ defined over $V$. For each prime l we then consider the galois representation on the $l$-torsion of $A_t$, where $t$ is a $K$-rational point of $V$. The largest possible image is $GSp_{2g}(\mathbb{F}_l)$ and in the cases $g=1$ and 2, we are able to get surjectivity for all $l$ and almost all $t$. In the case $g=1$ this recovers a theorem originally proven by William Duke.
We describe the relations among the $\ell$-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the $\ell$-torsion conjecture for $\ell$-groups and the other two conjectures for nilpotent groups.
Let K be a complete discrete valuation field of mixed characteristic (0,p) with perfect residue field. Let (π_n)_{n\ge 0} be a system of p-power roots of a uniformizer π=π_0 of K with π^p_{n+1}=π_n, and define G_s (resp.\ G_{\infty}) the absolute Galois group of K(π_s) (resp.\ K_{\infty}:=\bigcup_{n\ge 0} K(π_n)). In this paper, we study G_s-equivatiantness properties of G_{\infty}-equivariant homomorphisms between torsion (potentially) crystalline representations
We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31.
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$ without CM. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the correspond…
In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question essentially asks whether, up to isogeny, a Drinfeld module $ϕ$ over $\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of $ϕ$ at almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is $2$, but negative if the rank…
Let $K$ be a complete discrete valuation field of characteristic $(0, p)$ with perfect residue field, and let $T$ be an integral $\mathbb{Z}_p$-representation of $\mathrm{Gal}(\overline{K}/K)$. A theorem of T. Liu says that if $T/p^n T$ is torsion semi-stable (resp. crystalline) of uniformly bounded Hodge-Tate weights for all $n \geq 1$, then $T$ is also semi-stable (resp. crystalline). In this note, we show that we can relax the condition of "uniformly bounded Hodge-Tate weights" to an unbounded (log-)growth condition.
For each nonsingular hyperelliptic curve of arbitrary genus, we construct a natural injection from the Galois cohomology of 2-torsion subgroups of Jacobian varieties of the curve to the set of isomorphism classes of nonsingular complete intersections of two quadrics. This gives a generalization of the result of Flynn and Skorobogatov.
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.
We give a uniform bound on the degree of the maximal torsion cosets for subvarieties of an abelian variety. The proof combines algebraic interpolation and a theorem of Serre on homotheties in the Galois representation associated to the torsion subgroup of an abelian variety.
Typos in the abstract have been corrected. Let $ρ_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $GL_2(\mathcal O/π^n)$. Here $\mathcal O$ is a ramified DVR with uniformiser $π$. If $ρ_n$ satisfies mild hypotheses we lift it to a characteristic zero $\mathcal O$-valued geometric weight two representation. The earlier methods could handle only the unramified case. We show that the deformation ring of a residual representation arising from a newform can be arranged to be a prescribed DVR provided we choose a suitable auxiliary level. We extend earlier pr…
We consider the inverse Galois problem over function fields of positive characteristic p, for example, the inverse Galois problem over the projective line. We describe a method to construct certain Galois covers of the projective line and other curves, which are ordinary in the sense that their Jacobian has maximal p-torsion. We do this by constructing Galois covers of ordinary semi-stable curves, and then deforming them into smooth Galois covers.
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…
The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.
Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z…
We settle the automorphism groups of curves appearing in a classification list of smooth plane curves with at least two Galois points. One of them is an ordinary curve whose automorphism group exceeds the Hurwitz bound.
We show that the generalized Giulietti-Korchmáros curve is not a Galois subcover of the Hermitian curve over $\mathbb{F}_{q^{2n}}$. This answers a question raised by Garcia, Güneri and Stichtenoth.
We determine conditions for the invariance of the gonality under base extension, depending on the numeric invariants of the curve. More generally, we study the Galois descent of morphisms of curves to Brauer-Severi varieties, and also of rational normal scrolls.
What is the true order of growth of torsion in the cohomology of an arithmetic group? Let $D$ be a quaternion over an imaginary quadratic field $F.$ Let $E/F$ be a cyclic Galois extension with $\mathrm{Gal}(E/F) = \langle σ\rangle.$ We prove lower bounds for "the Lefschetz number of $σ$ acting on torsion cohomology" of certain Galois-stable arithmetic subgroups of $D_E^\times.$ For these same subgroups, we unconditionally prove a would-be-numerical consequence of the existence of a hypothetical base change map for torsion cohomology.
We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.
Given an elliptic curve $E$ over a global function field $K$, the Galois action on the $n$-torsion points of $E$ gives rise to a mod-n Galois representation $ρ_{E,n}$. For $K$ satisfying some mild conditions, we show that the set of $E$ for which $ρ_{E,n}$ is as large as possible for all $n$, has density $1$.
The V_4-lines for each linearly normal space elliptic curve form the edges of a tetrahedron, however in case the elliptic curve has j=12^3, there exist Z_4-lines in addition. We show the arrangement of V_4 and Z_4-lines explicitly for the curve. As a corollary we obtain that each irreducible quartic curve with genus one has at most two Galois points.
In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the …
In 2017 Skabelund constructed two new examples of maximal curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti-Korchmáros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the Skabelund curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ is given. Calculating the genera of the corresponding curves, we find new additions to the list of known genera of maximal curves over finite fields.
In this paper we investigate the problem of lifting of Galois covers between algebraic curves from characteristic p>0 to characteristic 0. We prove a refined version of the main result of Garuti concerning this problem in [Ga]. We formulate a refined version of the Oort conjecture on liftings of cyclic Galois covers between curves. We introduce the notion of fake liftings of cyclic Galois covers between curves, their existence would contradict the Oort conjecture, and we study the geometry of their semi-stable models. Finally, we introduce and investigate on some examples the smoothening proce…
We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Ruetsche, respectively. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.
Let $G$ be a finite group and let $N/E$ be a tamely ramified $G$-Galois extension of number fields. We show how Stickelberger's factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic $\mathbb{Z}[G]$-modules attached to $N/E$. If $\mathcal{O}_N$ and $\mathcal{O}_E$ denote the rings of integers of $N$ and $E$ respectively, we get in particular that $\mathcal{O}_N\otimes_{\mathcal{O}_E}\mathcal{O}_N$ defines the trivial class in the class group $\mathrm{Cl}(\mathbb{Z}[G])$ and, if $N/E$ is also assumed to be locally abelian, that the square root of…
For any elliptic curve E defined over the rationals with complex multiplication and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of definition of torsion points of these curves, in particular we classify all cases where there are torsion points over Galois number fields not containing the field of definition of the CM.
Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.
We show how to speed up the computation of isomorphisms of hyperelliptic curves by using covariants. We also obtain new theoretical and practical results concerning models of these curves over their field of moduli.
Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden.
Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.
