Large families of elliptic curves ordered by conductor

2021 ◽  
Vol 157 (7) ◽  
pp. 1538-1583
Author(s):  
Ananth N. Shankar ◽  
Arul Shankar ◽  
Xiaoheng Wang
Keyword(s):  
Elliptic Curves ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Good Reduction ◽  
Positive Proportion ◽  
The Family ◽  
Large Families ◽  
Average Size

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$ . Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$ -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$ . The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

scholarly journals The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

2019 ◽  
Vol 101 (1) ◽  
pp. 299-327 ◽  
Author(s):  
Manjul Bhargava ◽  
Noam Elkies ◽  
Ari Shnidman
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Average Size

scholarly journals Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

2016 ◽  
Vol 19 (A) ◽  
pp. 351-370 ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Wei Ho ◽  
Nathan Kaplan ◽  
Simon Spicer ◽  
William Stein ◽  
...  
Keyword(s):  
Elliptic Curves ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Selmer Group ◽  
Average Rank ◽  
Recent Developments ◽  
Theoretical Results

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$, ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$, ordered by height, in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

scholarly journals Pseudoprime Reductions of Elliptic Curves

2012 ◽  
Vol 64 (1) ◽  
pp. 81-101 ◽  
Author(s):  
C. David ◽  
J. Wu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Complex Multiplication ◽  
Upper Bounds ◽  
Good Reduction ◽  
Elliptic Pseudoprimes

Abstract Let E be an elliptic curve over ℚ without complex multiplication, and for each prime p of good reduction, let nE(p) = |E(𝔽p)|. For any integer b, we consider elliptic pseudoprimes to the base b. More precisely, let QE,b(x) be the number of primes p ⩽ x such that bnE(p) ≡ b (mod nE(p)), and let πpseuE,b (x) be the number of compositive nE(p) such that bnE(p) ≡ b (mod nE(p)) (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for QE,b(x) and πpseuE,b (x), generalising some of the literature for the classical pseudoprimes to this new setting.

scholarly journals Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

2021 ◽  
pp. 1-28
Author(s):  
Antonio Lei ◽  
Meng Fai Lim
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Finite Extension ◽  
Selmer Groups ◽  
Good Reduction ◽  
Cyclotomic Extensions ◽  
Iwasawa Algebra

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all primes above [Formula: see text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic [Formula: see text]-extension of a finite extension [Formula: see text] of [Formula: see text] where [Formula: see text] is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell–Weil ranks of [Formula: see text] over a subextension of the cyclotomic [Formula: see text]-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the [Formula: see text]-parts of the Tate–Shafarevich groups of [Formula: see text] over these extensions.

scholarly journals Rational points on hyperelliptic curves having a marked non-Weierstrass point

2017 ◽  
Vol 154 (1) ◽  
pp. 188-222
Author(s):  
Arul Shankar ◽  
Xiaoheng Wang
Keyword(s):  
Weierstrass Point ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Selmer Groups ◽  
Automorphic Representations ◽  
The Family ◽  
Odd Degree ◽  
Fundamental Research ◽  
Average Size ◽  
Fixed Genus

In this paper, we consider the family of hyperelliptic curves over$\mathbb{Q}$having a fixed genus$n$and a marked rational non-Weierstrass point. We show that when$n\geqslant 9$, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as$n$tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of$5/2$on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2)180(2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, inAutomorphic representations and$L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

scholarly journals A Stability Result for the Union-Closed Size Problem

2015 ◽  
Vol 25 (3) ◽  
pp. 399-418
Author(s):  
TOM ECCLES
Keyword(s):  
Positive Constant ◽  
Stability Result ◽  
Size Problem ◽  
The Family ◽  
Large Families ◽  
Average Size

A family of sets is called union-closed if whenever A and B are sets of the family, so is A ∪ B. The long-standing union-closed conjecture states that if a family of subsets of [n] is union-closed, some element appears in at least half the sets of the family. A natural weakening is that the union-closed conjecture holds for large families, that is, families consisting of at least p02n sets for some constant p0. The first result in this direction appears in a recent paper of Balla, Bollobás and Eccles [1], who showed that union-closed families of at least $\tfrac{2}{3}$2n sets satisfy the conjecture; they proved this by determining the minimum possible average size of a set in a union-closed family of given size. However, the methods used in that paper cannot prove a better constant than $\tfrac{2}{3}$. Here, we provide a stability result for the main theorem of [1], and as a consequence we prove the union-closed conjecture for families of at least ($\tfrac{2}{3}$ − c)2n sets, for a positive constant c.

scholarly journals Average size of 2-Selmer groups of elliptic curves over function fields

2014 ◽  
Vol 21 (6) ◽  
pp. 1305-1339
Author(s):  
Q.P. Hô ◽  
V.B. Lê Hùng ◽  
B.C. Ngô
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Selmer Groups ◽  
Average Size
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