We show that two ordinary isogenous elliptic curves have isomorphic groups of rational points if they have the same $j$-invariant and we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 or 1728. Using a result by Heuberger and Mazzoli we establish a general case of this relationship within isogenous elliptic curves not necessarily having equal $j$-invariant.
Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where $K$ is an $\ell$-power extension of $k$. In a previous work we have shown that, under mild rationality hypotheses, the case of interest is when $\ell=2$ and $K$ is the unique quadratic extension of $k$. In this paper we study the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E'$ over ${\mathbf{Q}}$ and asking for the …
We prove that the family $\mathcal{I}_{F_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with $F_0$-rational $j$-invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family $\mathcal{I}_{d_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree $d_0$ $j$-invariant is typically bounded in torsion.
Given a pair of abelian varieties defined over a number field k and isogenous over a finite Galois extension L/k, we define a rational Artin representation of the group Gal(L/k) that shows a global relation between the L-functions of each variety and provides certain information about their decomposition up to isogeny over L. We study several properties of these Artin representations. As an application, for each curve C' in a family of twists of a certain genus 3 curve C, we explicitly compute the Artin representation attached to the Jacobians of C and C' and show how this Artin representation…
Faltings's isogeny theorem states that two abelian varieties are isogenous over a number field precisely when the characteristic polynomials of the reductions at almost all prime ideals of the number field agree. This implies that two abelian varieties over $\mathbb{Q}$ with the same $L$-series are necessarily isogenous, but this is false over a general number field. Let $A$ and $A'$ be two abelian varieties, defined over number fields $K$ and $K'$ respectively. Our main result is that $A$ and $A'$ are isogenous after a suitable isomorphism between $K$ and $K'$ if and only if the Dirichlet cha…
We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C' are curves over a finite field K, with K-rational base points P and P', and let D and D' be the pullbacks (via the Abel-Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C,P) and (C',P') are *doubly isogenous* if Jac(C) and Jac(C') are isogenous over K and Jac(D) and Jac(D') are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the nu…
Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is …
Elliptic $K$-curves are elliptic curves defined over some field extension $L/K$ that are isogenous to all of their Galois conjugates. We present a new result on $K$-curves $E$ that are given by a $K$-rational orbit $\{τ, -1/Nτ\}$ of the Fricke involution on $Y_0(N)$, giving a simple Diophantine condition on the extension $L/K$ that determines which twists of $E$ allow the isogeny between Galois conjugates to be defined over $L$. To support and illustrate this result, we also discuss parameterizations of cyclic $N$-isogenies corresponding to points on modular curves $X_0(N)$ of genus $0$. These…
We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose 9-torsion subgroups are isomorphic as Galois modules.
We work out the complete descent via 4-isogeny for a family of rational elliptic curves with a rational point of order 4; such a family is of the form $y^2 + x y + a y = x^3 + a x^2$ where $\sqrt{-a} \in \mathbb Q^\times$. In the process we exhibit the 4-isogeny and the isogenous curve, explicitly present the principal homogeneous spaces, and discuss examples by computing the rank.
We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.
We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose n-torsion subgroups are isomorphic as Galois modules. We also show by giving explicit non-trivial examples over Q(T) that there are infinitely many examples over Q in the cases n=9 and n=11.
Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$. We compute certain twists of the classical modular curves $X(8)$. Searching for rational points on these twists enables us to find non-trivial pairs of $8$-congruent elliptic curves over $\mathbb{Q}$, i.e. pairs of non-isogenous elliptic curves over $\mathbb{Q}$ whose $8$-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over $\mathbb{Q}$.
We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin's unity, using the presentation of the Kummer surfaces as isotrivial elliptic fibration and as double cover of the modular elliptic surface of level two. By carrying out the rational point-count with respect to either of the two elliptic fibrations explicitly, we obtain an interesting new identity between two-parameter counting functions.
In the previous paper, Hirakawa and the author determined the set of rational points of a certain infinite family of hyperelliptic curves $C^{(p;i,j)}$ parametrized by a prime number $p$ and integers $i$, $j$. In the proof, we used the standard $2$-descent argument and a Lutz-Nagell theorem that was proven by Grant. In this paper, we extend the above work. By using the descent theorem, the proof for $j=2$ is reduced to elliptic curves of rank $0$ that are independent of $p$. On the other hand, for odd $j$, we consider another hyperelliptic curve $C'^{(p;i,j)}$ whose Jacobian variety is isogeno…
We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$ is the modular curve $X_1(N)$, most known examples of sporadic points come from elliptic curves with complex multiplication (CM). We seek to understand all sporadic points on $X_1(N)$ corresponding to $\mathbb{Q}$-curves, which are elliptic curves isogenous to their Galois conjugates. This class contains not only all CM elliptic curves, but also any elliptic curve $\overline{\mathbb{Q}}$-isogenous to one with a rational $j$-invariant, among others…
Consider a pair of ordinary elliptic curves $E$ and $E'$ defined over the same finite field $\mathbb{F}_q$. Suppose they have the same number of $\mathbb{F}_q$-rational points, i.e. $|E(\mathbb{F}_q)|=|E'(\mathbb{F}_q)|$. In this paper we characterise for which finite field extensions $\mathbb{F}_{q^k}$, $k\geq 1$ (if any) the corresponding groups of $\mathbb{F}_{q^k}$-rational points are isomorphic, i.e. $E(\mathbb{F}_{q^k}) \cong E'(\mathbb{F}_{q^k})$.
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $p \neq \ell$, the Frobenius automorphism associated to $p$ (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of $\ell$. Its splitting field is called the Frobenius field of $E$ at $p$. Let $E_1$ and $E_2$ be two elliptic curves defined over $\mathbb{Q}$ that are non-isogenous over $\overline{\mathbb{Q}}$ and also without complex multiplication over $\…
Let $k$ be a number field. We give an explicit bound, depending only on $[k:\mathbf{Q}]$ and the discriminant of the Néron--Severi lattice, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on $[k:\mathbf{Q}]$ and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, …
Let $n>1$ be an integer such that $X_{0}\!\left( n\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over $\mathbb{Q}$. We achieve this by introducing $56$ parameterized families of elliptic curves $\mathcal{C}_{n,i}(t,d)$ defined over $K(t,d)$, which have the following two properties for a fixed $n$: the elliptic curves $\mathcal{C}_{n,i}(t,d)$ are isogenous over $K(t,d)$, and there are integers $k_{1}$ and $k_{2}$ such tha…
We compute twists of the modular curve $X(13)$ that parametrise the elliptic curves 13-congruent to a given elliptic curve. Searching for rational points on these twists enables us to find non-trivial pairs of 13-congruent elliptic curves over ${\mathbb Q}$, i.e. pairs of non-isogenous elliptic curves over ${\mathbb Q}$ whose 13-torsion subgroups are isomorphic as Galois modules. We also find equations for the surfaces parametrising pairs of 13-congruent elliptic curves. There are two such surfaces, corresponding to 13-congruences that do, or do not, respect the Weil pairing. We write each as …
For a semi-abelian variety over a global function field which is isogenous to an isotrivial one, we show that on the product of local points of a subvariety satisfying a minor condition, the topological closure of a finitely generated subgroup of %the group of its global points cuts out exactly the global points of the subvariety lying in this subgroup. As a corollary, on every non-isotrivial super-singular curve of genus two over a global function field, we conclude that the Brauer-Manin condition cuts out exactly the set of its rational points.
We consider elliptic curves over global fields of positive characteristic with two distinct marked non-trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of these curves exists, in a natural sense, and equals 12. Along the way, we consider a map from these 2-Selmer groups to the moduli space of $G$-torsors over an algebraic curve, where $G$ is isogenous to $\mathrm{SL}_2^4$, and show that the images of 2-Selmer elements under this map become equidistributed in the limit.
For each odd prime $p$, we prove the existence of infinitely many real quadratic fields which are $p$-rational. Explicit imaginary and real bi-quadratic $p$-rational fields are also given for each prime $p$. Using a recent method developed by Greenberg, we deduce the existence of Galois extensions of $\mathbf{Q}$ with Galois group isomorphic to an open subgroup of $GL_n(\mathbf{Z_p})$, for $n =4$ and $n =5$ and at least for all the primes $p <192.699.943$.
Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null…
In this paper, we define a new product over $\mathbb{R}^{\infty}$, which allows us to obtain a group isomorphic to $\mathbb R^*$ with the usual product. This operation unexpectedly offers an interpretation of the Rédei rational functions, making more clear some of their properties, and leads to another product, which generates a group structure over the Pell hyperbola. Finally, we join together these results, in order to evaluate solutions of Pell equation in an original way.
In this paper, we obtain two analogues of the Sturm bound for modular forms in the function field setting. In the case of mixed characteristic, we prove that any harmonic cochain is uniquely determined by an explicit finite number of its first Fourier coefficients where our bound is much smaller than the ones in the literature. A similar bound is derived for generators of the Hecke algebra on harmonic cochains. As an application, we present a computational criterion for checking whether two elliptic curves over the rational function field $\mathbb{F}_q(θ)$ with same conductor are isogenous. In…
Weyl group multiple Dirichlet series are Dirichlet series in $r$ complex variables, with analytic continuation to $\mathbb{C}^r$ and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank $r$. Such series may be defined for any global field $K$, but in the case when $K$ is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when $K=\mathbb{F}_q(T)$, and describe the denominators and support of the numerators.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d\in\dbN$. Let $b_{c,d}\ge 1$ be the smallest integer such that for any two $p$-divisible groups $H$ and $H^\prime$ over $k$ of codimension $c$ and dimension $d$ the following assertion holds: If $H[p^{b_{c,d}}]$ and $H^\prime[p^{b_{c,d}}]$ are isomorphic, then $H$ and $H^\prime$ are isogenous. We show that $b_{c,d}=\lceil{cd\over {c+d}}\rceil$. This proves Traverso's isogeny conjecture for $p$-divisible groups over $k$.
Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^\infty)$ be the compositum of all extensions of $\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\mathbb{Q}(D_4^\infty)$ is in fact the compositum of all $D_4$ extensions of $\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^\infty))$ is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have…
An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines…
To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Kuwata. We give an explicit description of the geometric Mordell-Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the non-…
Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of these involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $\mathbb{C}$ …
We show that a given rational Shimura curve Y with strictly maximal Higgs field in the moduli space of g-dimensional abelian varieties does not generically intersect the Schottky locus for large g. We achieve this by using a result of Viehweg and Zuo which says that if Y parameterizes a family of curves of genus g, then the corresponding family of Jacobians is isogenous over Y to the g-fold product of a modular family of elliptic curves. After reducing the situation from the field of complex numbers to a finite field, we will see, combining the Weil and Sato-Tate conjectures, that this is impo…
The set of morphisms $\f:\PP^1\to\PP^1$ of degree $d$ is parametrized by an affine open subset $\Rat_d$ of $\PP^{2d+1}$. We consider the action of~$\SL_2$ on $\Rat_d$ induced by the {\it conjugation action\/} of $\SL_2$ on rational maps; that is, $f\in\SL_2$ acts on~$\f$ via $\f^f=f^{-1}\circ\f\circ f$. The quotient space $\M_d=\Rat_d/\SL_2$ arises very naturally in the study of discrete dynamical systems on~$\PP^1$. We prove that~$\M_d$ exists as an affine integral scheme over~$\ZZ$, that $\M_2$ is isomorphic to~$Å^2_\ZZ$, and that the natural completion of~$\M_2$ obtained using geometric inv…
Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite sequences of explicit elliptic curves $E$, for which we prove that their Tate-Shafarevich group $\mathrm{III}(E)$ is finite and satisfies $|\mathrm{III}(E)| = H(E)^{1+o(1)}$ as $H(E)\to\infty$, where $H(E)$ denotes the exponential differential height of $E$. The elliptic curves in these sequences are pairwise neither isogenous nor geometrically isomorphic. We furthe…
Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet L-series of the two curves are equal via an isomorphism of their Dirichlet character group…
Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite characteristic (G. Cornelissen); Flat connections and representations of the fundamental group in characteristic p>0 (C. Gasbarri) Questions on p-torsion of hyperelliptic curves (D. Glass & R. Pries); Automorphisms of curves and stable reduction (C. Lehr & M. Matignon); p-groups as automorphism groups of curves in characteristic p; Lifting Galois covers of smooth curv…
For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group of $X_0(N)$, which is conjecturally equal to the rational torsion subgroup of $J_0(N)$. More specifically, for a given prime $\ell$, we construct a rational cuspidal divisor $Z_\ell(d)$ for any non-trivial divisor $d$ of $N$. Also, we compute the order of the linear equivalence class of the divisor $Z_\ell(d)$ and show that the $\ell$-primary subgroup of the rational cuspidal divisor class group of $X_0(N)$ is isomorphic to the direct sum of the cyclic subgroups generated by the lin…
Let G be a finite group and V a finite-dimensional rational G-representation. We ask whether there exists a finite Galois extension L/K of number fields with Galois group G, an elliptic curve E/K, and a G-submodule of E(L) tensor Q isomorphic to V.
Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in $S$, we give a complete description of triples $(f_1,f_2,p)$ such that $p$ is a rational periodic point for both $f_i\in S$, $i=1,2$. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two n…
Let $G$ be a split reductive group over a finite field $\Fq$. Let $F=\Fq(t)$ and let $\A$ denote the adèles of $F$. We show that every double coset in $G(F)\bsl G(\A)/ K$ has a representative in a maximal split torus of $G$. Here $K$ is the set of integral adèlic points of $G$. When $G$ ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.
