Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups
نویسندگان
چکیده
We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|{ш }(E)| \gt 63408^2$, which was largest previously known value for any explicit curve. Our record is an curve $E$ = 1029212^2 2^4\cdot 79^2 \cd
منابع مشابه
Elliptic curves with large analytic order of the Tate-Shafarevich group
Let E be an elliptic curve over Q of conductor N = N(E) and let L(E, s) = ∑∞ n=1 ann −s denote the corresponding L-series. Let X(E) be the Tate-Shafarevich group of E (conjecturally finite), E(Q) be the group of global points, and R be the regulator (with respect to the Néron-Tate height pairing). Finally, let ω be the real period, and c∞ = ω or 2ω (according to whether E(R) is connected or not...
متن کاملThe analytic order of III for modular elliptic curves
In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny ...
متن کاملAnalytic Problems for Elliptic Curves
We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an intere...
متن کاملOn prime-order elliptic curves with embedding
We further analyze the solutions to the Diophantine equations from which prime-order elliptic curves of embedding degrees k = 3, 4 or 6 (MNT curves) may be obtained. We give an explicit algorithm to generate such curves. We derive a heuristic lower bound for the number E(z) of MNT curves with k = 6 and discriminant D ≤ z, and compare this lower bound with experimental data.
متن کاملComputing in Component Groups of Elliptic Curves
Let K be a p-adic local field and E an elliptic curve defined over K. The component group of E is the group E(K)/E0(K), where E0(K) denotes the subgroup of points of good reduction; this is known to be finite, cyclic if E has multiplicative reduction, and of order at most 4 if E has additive reduction. We show how to compute explicitly an isomorphism E(K)/E0(K) ∼= Z/NZ or E(K)/E0(K) ∼= Z/2Z× Z/2Z.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Colloquium Mathematicum
سال: 2021
ISSN: ['0010-1354', '1730-6302']
DOI: https://doi.org/10.4064/cm8008-9-2020