Selmer Ranks Research Articles

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p=2 obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL(2): At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families.

Erratum to "Disparity in Selmer ranks of quadratic twists of elliptic curves"

Erratum to "Disparity in Selmer ranks of quadratic twists of elliptic curves"

On the Average of p-Selmer Ranks in Quadratic Twist Families of Elliptic Curves Over Global Function Fields

Abstract Let $\mathbb{F}_{q}$ be a finite field whose characteristic is relatively prime to $2$ and $3$. Let $p$ be a prime number that is coprime to $q$. Let $E$ be an elliptic curve over the global function field $K = \mathbb{F}_{q}(t)$ such that $\textrm{Gal}(K(E[p])/K)$ contains the special linear group $\textrm{SL}_{2}(\mathbb{F}_{p})$. We show that if the quadratic twist family of $E$ has an element whose Néron model has a multiplicative reduction away from $\infty $, then the average $p$-Selmer rank is $p+1$ in large $q$-limit for almost all primes $p$.

On the extension of even families of non-congruent numbers

A method that extends existing families of even non-congruent numbers to produce new families of non-congruent numbers with arbitrarily many distinct prime factors is presented. We show that infinitely many new non-congruent numbers can be generated by appending a suitable collection of primes onto any even non-congruent number whose corresponding congruent number elliptic curve has 2-Selmer rank of zero. Our method relies upon Monsky’s formula for computing the 2-Selmer rank of the congruent number elliptic curve. Even non-congruent numbers constructed according to our result have an unlimited number of prime factors in each odd congruence class modulo eight, and have congruent number elliptic curves with 2-Selmer rank equal to zero.

Construction of an infinite family of elliptic curves of 2-Selmer rank 1 from Heron triangles

Given any positive integer n, there exist triangles, called Heron triangles, with rational sides whose area is n. Assuming the finiteness of the Shafarevich–Tate group, we construct a family of infinitely many Heronian elliptic curves of rank 1 arising from Heron triangles of a certain type. We then explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.

Bounds for 2-Selmer ranks in terms of seminarrow class groups

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]/{(F(x))}$, which we call the \textit{semi-narrow class group} of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E/K$ is semistable and the discriminant of $F$ is totally negative, then we frequently determine the $2$-Selmer rank of $E$ by computing the root number of $E$ and the $2$-rank of the narrow class group of $L$.

Distribution of the 2-Selmer rank under twisting

Distribution of the 2-Selmer rank under twisting

Families of non-congruent numbers with odd prime factors of the form 8k + 3

A congruent number is a positive integer which can be represented as the area of a right triangle such that all of its side lengths are rational numbers. The problem determining whether a given number is congruent is usually studied by computing the Mordell-Weil rank of the corresponding elliptic curve. The Monsky matrix gives a way to compute efficiently the 2-Selmer rank, thereby gives an upper bound for the Mordell-Weil rank. In this paper, by using Monsky's matrix, we present new families of non-congruent numbers such that all of their odd prime factors are of the form 8k+3. Our result generalizes previous works of Reinholz–Spearman–Yang [12] and Cheng–Guo [3].

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields K K (the trivial bound being O ϵ , n ( | D i s c ( K ) | 1 / 2 + ϵ ) O_{\epsilon ,n}(|\mathrm {Disc}(K)|^{1/2+\epsilon }) coming from the bound on the entire class group). This yields corresponding improvements to: (1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves, (2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves, (3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves, and (4) bounds of Baily and Wong on the number of A 4 A_4 -quartic fields of bounded discriminant.

On conjectural rank parities of quartic and sextic twists of elliptic curves

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 [Formula: see text]-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 field, after a finite calculation. This generalizes the previous results of Hadian and Weidner on the behavior of [Formula: see text]-Selmer ranks under [Formula: see text]-twists.

Quadratic twists of abelian varieties and disparity in Selmer ranks

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree and we describe both.

The non-congruent numbers via Monsky’s formula

This paper gives some new families of non-congruent numbers with arbitrarily many prime divisors. The main idea is based on Monsky’s formula for the 2-Selmer rank of congruent elliptic curves.

Some new families of non-congruent numbers

We construct several new families of non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8. Our results generalize the work of Reinholz, Spearman and Yang [14]. Our methods are based on Monsky's formula on the 2-Selmer rank of the congruent elliptic curve and the recent results by Tian, Yuan and Zhang on the congruent number problem.

$2$-Selmer near-companion curves

Let $E$ and $A$ be elliptic curves over a number field $K$. Let $\chi$ be a quadratic character of $K$. We prove the conjecture posed by Mazur and Rubin on $n$-Selmer near-companion curves in the case $n=2$. Namely, we show if the difference of the $2$-Selmer ranks of $E^\chi$ and $A^\chi$ is bounded independent of $\chi$, there is a $G_K$-isomorphism $E[2] \cong A[2]$.

On $2$-Selmer ranks of quadratic twists of elliptic curves

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 $t_E$.

Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion

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 about the minimum number of twists of E with rank 0, and subject to standard conjectures, the number of twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K.

Level raising mod 2 and arbitrary 2-Selmer ranks

We prove a level raising mod $\ell =2$ theorem for elliptic curves over $\mathbb{Q}$. It generalizes theorems of Ribet and Diamond–Taylor and also explains different sign phenomena compared to odd $\ell$. We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.

Selmer ranks of twists of hyperelliptic curves and superelliptic curves

We study the variation of Selmer ranks of Jacobians of twists of hyperelliptic curves and superelliptic curves. We find sufficient conditions for such curves to have infinitely many twists whose Jacobians have Selmer ranks equal to r, for any given nonnegative integer r. This generalizes earlier results of Mazur–Rubin on elliptic curves.

THE DISTRIBUTION OF 2‐SELMER RANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO‐TORSION

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K 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 or full rational two-torsion.

A Markov model for Selmer ranks in families of twists

AbstractWe study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.