Let p be a prime and K be a number field. Let rho_{E,p}:G_K \longrightarrow Aut(T_p E)\cong GL_2(Z_p) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the image of rho_{E,p} is open if E has no complex multiplication. For an elliptic curve E over K whose j-invariant does not appear in an exceptional finite set, we give an explicit uniform lower bound of the size of the image of rho_{E,p}.
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.
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.
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 local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.
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 E/Q be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of Q obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group H^1(G, E[p]) does not vanish, and investigate the analogous question for E[p^i] when i>1. We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald-Wang problem for elliptic curves.
Let $K$ be a $p$-adic field and $E$ an elliptic curve over $K$ with potential good reduction. For some large Galois extensions $L$ of $K$ containing all $p$-power roots of unity, we show the vanishing of certain Galois cohomology groups of $L$ with values in the $p$-adic representation associated with $E$. This generalizes some results of Coates, Sujatha and Wintenberger.
We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing $ζ_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic fields $\mathbb{Q}(ζ_n)$ with $5\nmid n$, this proportion is at least $1-\varepsilon$ with only finitely many exceptions of $n$, for any choice of $\varepsilon > 0$.
For $\ell = 3$ and 5 it is known that every odd, irreducible, 2-dimensional representation of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ and determinant equal to the cyclotomic character must "come from" the $\ell$-torsion points of an elliptic curve defined over $\Q$. We prove, by giving concrete counter-examples, that this result is false for every prime $\ell >5$.
We show how rational points on certain varieties parametrize phenomena arising in the Galois theory of iterates of quadratic polynomials. As an example, we characterize completely the set of quadratic polynomials $x^2+c$ whose third iterate has a "small" Galois group by determining the rational points on some elliptic curves. It follows as a corollary that the only such integer value with this property is $c=3$, answering a question of Rafe Jones. Furthermore, using a result of Granville's on the rational points on quadratic twists of a hyperelliptic curve, we indicate how the ABC conjecture i…
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…
In this paper we present a description of the Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and $SL_2(\mathbb{F}_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over $\mathbb{Q}_2$ is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.
Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $ρ_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$ that describes the Galois action on the $\ell$-torsion points of $E$. Building on recent work of Rouse--Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over $\mathbf{Q}$ with simultaneously non-surjective, composite image of Galois. We also provably determine the rational points on almost …
In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective representations $ρ$ of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic curve defined over $\Q$, but not minimally, i.e., so that any elliptic curve giving rise to $ρ$ has prime-to-$\ell$ conductor greater than the (prime-to-$\ell$) conductor of $ρ$. In this brief note, we will show that the same is true for any prime $\ell >7$, concretely, we will show that for any such $\ell$ the elliptic curve $$E^\ell: \qquad Y^2 = X (X- 3^\ell ) …
Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplif…
We give a classification of all possible $2$-adic images of Galois representations associated to elliptic curves over $\mathbb{Q}$. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.
The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let $ρ_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\t…
We give a quantitative version of a result due to N. Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's Theorem states that, on average over a suitably chosen algebraic family, the L-function of an elliptic curve over a function field becomes "as irreducible as possible" when seen as a polynomial with rational coefficients, as the cardinality of the field of constants grows. A quantitative refinement is obtained as a corollary of our main result which gives an estimate for the proportion of elliptic curves studied whose L-functions have …
We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has Galois group permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups $G$, we show that most elliptic curves over (e.g.) $\mathbb{Q}$ gain rank over infinitely many $G$-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion which allows to deduce that "many" elliptic curves gain rank over infin…
Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the number of elliptic curves over ${\bf Q}$ with at least $i$ pairs of Galois-stable cyclic subgroups of order 4, and height at most $X$. In this article we show that $N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$. We also show, as $X\to \infty$, that $N_2(X)=c_{2,1}X^{1/6}+o(X^{1/12})$, the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann ze…
An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…
Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.
We study the growth of the Galois invariants of the $p$-Selmer group of an elliptic curve in a degree $p$ Galois extension. We show that this growth is determined by certain local cohomology groups and determine necessary and sufficient conditions for these groups to be trivial. Under certain hypotheses this allows us to give necessary and sufficient conditions for there to be growth in the full $p$-Selmer group in a degree $p$ Galois extension.
We prove useful necessary and sufficient conditions for an elliptic curve over a number field to admit a surjective adelic Galois representation. Using these conditions, we compute an example of a number field K and an elliptic curve E/K whose corresponding adelic representation is surjective.
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…
We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.
In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is totally inert in $K$ and $E$ any elliptic curve defined over $K$ such that $E$ has potentially multiplicative reduction at the prime $\mathfrak q$ above $q$. Then for every rational prime $p> B_d$, $E$ has an irreducible mod $p$ Galois representation. This result has Diophantine applications within the "modular method". We present one such application in the form of an Asymp…
This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations with difference Galois group $\operatorname{GL}_{2}(\mathbb C)$.
Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$ is surjective for all but finitely many $\ell$. We say that a prime number $\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particular…
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.
For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $ρ_E : G \to GL_2(\hat{\mathbb{Z}})$, where $G$ is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of $ρ_E$ is open and hence has finite index in $GL_2(\hat{\mathbb{Z}})$. We will study what indices are possible assuming that we are willing to exclude a finite number of possible $j$-invariants from consideration. For example, we will show that there is a finite set $J$ of rational numbers such that if $E/\…
Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $α\in E(F)$ of infinite order. Attached to the pair $(E,α)$ is the $\ell$-adic arboreal Galois representation $ω_{E,α,\ell^{\infty}} : {\rm Gal}(\overline{F}/F) \to \mathbb{Z}_{\ell}^{2} \rtimes {\rm GL}_{2}(\mathbb{Z}_{\ell})$ describing the action of ${\rm Gal}(\overline{F}/F)$ on points $β_{n}$ so that $\ell^{n} β_{n} = α$. We give an explicit bound on the index of the image of $ω_{E,α,\ell^{\infty}}$ depending on how $\ell$-divisible the point $α$ is, and the image of the ordinary $\el…
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 …
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.
In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and Tamagawa, related with the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion.
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.
We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.
Let $K$ be a number field and let $E/K$ be an elliptic curve whose mod $\ell$ Galois representation locally has image contained in a group $G$, up to conjugacy. We classify the possible images for the global Galois representation in the case where $G$ is a Cartan subgroup or the normalizer of a Cartan subgroup. When $K = \mathbf{Q}$, we deduce a counterexample to the local-global principle in the case where $G$ is the normalizer of a split Cartan and $\ell = 13$. In particular, there are at least three elliptic curves (up to twist) over $\mathbf{Q}$ whose mod $13$ image of Galois is locally co…
We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage tree. Moreover, we show that the index is bounded, in most cases, independently of the polynomial.
Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre's open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. In this note, we prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.
Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus bigger than 1.
Building on ideas of Vatsal, Cornut proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's 2-dimensional l-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.
