A formula is given for the dimension of the Selmer group of the rational three-isogeny of elliptic curves of the form y^2=x^3+a(x-b)^2. The formula is in terms of the three-ranks of the quadratic number fields Q(\sqrt{a}) and Q(\sqrt{-3a}) and various aspects of the arithmetic of these number fields. In addition a duality theorem is used to relate the dimension of the Selmer group of the three-isogeny with the dimension of the Selmer group of its dual isogeny.
We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated $\mathbb{Z}_p$-module. The relationship between this conjecture and Iwasawa's classical $μ=0$ conjecture is clarified. We also present some partial results towards the question whether Conjecture A is invariant under isogenies.
We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.
The elliptic curve $E_k \colon y^2 = x^3 + k$ admits a natural 3-isogeny $ϕ_k \colon E_k \to E_{-27k}$. We compute the average size of the $ϕ_k$-Selmer group as $k$ varies over the integers. Unlike previous results of Bhargava and Shankar on $n$-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on $k$; this sensitivity can be precisely controlled by the Tamagawa numbers of $E_k$ and $E_{-27k}$. As consequences, we prove that the average rank of the curves $E_k$, $k\in\mathbb Z$, is less than 1.21 and over $23\%$ (resp. $41\%$) of the cur…
For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension one, we deduce that if $E/F$ is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded.…
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results abo…
We study the behavior under twisting of the Selmer rank parities of a self-dual prime-degree isogeny on a principally polarized abelian variety defined over a number field, subject to compatibility relations between the twists and the isogeny. In particular, we study isogenies on abelian varieties whose Selmer rank parities are related to the rank parities of elliptic curves with $j$-invariant 0 or 1728, assuming the Shafarevich-Tate conjecture. Using these results, we show how to classify the conjectural rank parities of all quartic or sextic twists of an elliptic curve defined over a number …
Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.
Let h be a p-isogeny of elliptic curves. We describe how to perform h-descents on the nontrivial elements in the Shafarevich-Tate group which are killed by the dual isogeny h'. This makes computation of p-Selmer groups of elliptic curves admitting a p-isogeny over Q feasible for p = 5,7 in cases where an isogeny descent is insufficient and a full p-descent would be infeasible. As an application we complete the verification of the full Birch and Swinnerton-Dyer conjectural formula for all elliptic curves over Q of rank zero or one and conductor less than 5000.
In this note, we compare the dual Selmer groups of an abelian variety with that of its dual over certain large Galois field. We give formula which relates the generalized Iwasawa $μ$-invariants associated with their dual Selmer groups under the natural isogeny.
In recent work, Bhargava and Shankar have shown that the average size of the $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ is exactly $3$, and Bhargava and Ho have shown that the average size of the $2$-Selmer group in the family of elliptic curves with a marked point is exactly $6$. In contrast to these results, we show that the average size of the $2$-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous…
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o…
In this paper, we study completely faithful torsion $\mathbb{Z}_p[[G]]$-modules with applications to the study of Selmer groups. Namely, if $G$ is a nonabelian group belonging to certain classes of polycyclic pro-$p$ group, we establish the abundance of faithful torsion $\mathbb{Z}_p[[G]]$-modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such $\mathbb{Z}_p[[G]]$-modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer gr…
We generalize a result of Frey [Fre88] on the Selmer group of twists of elliptic curves over Q with Q-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational point. We also provide examples of elliptic curves coming from [Zyw15] that satisfy the conditions of our Theorem B.
Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its pre-images into n orbits. It is shown that all such points above a certain height have their denominator divisible by at least n distinct primes. This generalizes Siegel's theorem and more recent results of Everest et al. For multiplication by a prime l, it is shown that if n>1 then either the point is l times a rational point or the elliptic curve emits a rational l-isogeny.
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.
On Conjectures of $μ$-Invariant for Selmer Groups at $p=2$
The University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Morningside Center of Mathematics（中国科学院数学与系统科学研究院；中国科学院数学与系统科学研究所）；Academy of Mathematics and Systems Sciences, Chinese Academy of Science
Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower bound on the number of D's for which the twists E(D) have 2-Selmer rank <= 1. We include in the introduction our (brief) opinion about why it is supposed to be hard to push our technique to make the Selmer group trivial.
We consider the family of hyperelliptic curves over $\Q$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also prove that the average size of the $ϕ$-Selmer groups of a family of isogenies associated to this family is exactly 2.
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 prove that the $2^\infty$-class groups of the imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra heuristic. Given an elliptic curve E/Q with full rational 2-torsion and no rational cyclic subgroup of order four, we analogously prove that the $2^\infty$-Selmer groups of the quadratic twists of E have distribution as predicted by Delaunay's heuristic. In particular, among the twists E^d with |d| < N, the number of curves with rank at least two is $o(N)$.
For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. Dokchitser have showed …
We show that if $E$ is an elliptic curve over $\mathbf{Q}$ with a $\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\mathbf{Q}(\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the…
We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.
We give new instances where Chabauty--Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the first explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell-Weil rank larger than its genus.
Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a totally complex field. We establish an algebraic function of the characteristic ideal of the Selmer group associated to Artin representation over the cyclotomic $\Z_p$- extension of the rational numbers under certain mild hypotheses and construct several examples to illustrate our result. We also prove that in this situation $μ$-invariant of the dual Selmer group is independent of the choice of the lattice.
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 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.
We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all integers larger than $c$ and having the same parity as $c$. We also find sufficient conditions on $A_E$ such that $A_E$ is equal to $\Z_{\ge t_E}$ for some number $t_E$. When all points in $E[2]$ are rational, we give an upper bound for $t_E$.
For an elliptic curve over the rational number field and a prime number $p$, we study the structure of the classical Selmer group of $p$-power torsion points. In our previous paper \cite{Ku6}, assuming the main conjecture and the non-degeneracy of the $p$-adic height pairing, we proved that the structure of the Selmer group with respect to $p$-power torsion points is determined by some analytic elements $\tildeδ_{m}$ defined from modular symbols. In this paper, we do not assume the main conjecture nor the non-degeneracy of the $p$-adic height pairing, and study the structure of Selmer groups, …
Given a rational point on a curve in a rational isogeny class, a natural question concerns the field of definition of its pre-images. The multiplication by m endomorphism is a powerful and much-used tool. The pre-images for this map are found by factorizing a monic polynomial of degree m^2. For m = 2, Everest and King gave examples where the existence of a quadratic factor coincided with the existence of a rational pre-image via a 2-isogeny. Nelson Stephens asked if this always happens and the question is answered in the affirmative. It is also shown that the analogue for m = 3 can only be fal…
Taking A to be an abelian variety with full 2-torsion over a number field k, we investigate how the 4-Selmer rank of the quadratic twist A^d changes with d. We show that this rank depends on the splitting behavior of the primes dividing d in a certain number field L/k. Assuming the grand Riemann hypothesis, we then prove that, given an elliptic curve E/Q with full rational 2-torsion, the quadratic twist family of E usually has the distribution of $4$-Selmer groups predicted by Delaunay's heuristic. Analogously, and still subject to the grand Riemann hypothesis, we prove that the set of quadrat…
Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p \geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_\infty/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group $G=\operatorname{Gal}(K_\infty/K)$. Under some mild additional hypotheses this gives an algebraic functional equation of the conjectured $p$-adic L-function. As an application we construct comp…
In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg representation of the group $G=\text{PSO}(2n+2)$ and a Hitchin fibration. Consistent with the result over Q of Arul Shankar and Xiaoheng Wang [Rational points on hyperelliptic curves having a marked non-Weierstrass point, in Compos. Math., 154(1):188-222, 2018], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperell…
