Rank Of Elliptic Curves Research Articles

On elliptic curves assigned to a pair of right triangles with an equal hypotenuse

Consider a pair of right triangles with an equal hypotenuse. This turns out to solve the diophantine system of equations a 2 + b 2 = c 2 + d 2 = e 2 in integers. To this system we associate a family of elliptic curves with defining equation y 2 = (x−a 2)(x−b 2)(x−c 2)+a 2 b 2 c 2. We show that there exists a subfamily of rank ≥ 3 over ℚ(m, n, k, ℓ) and obtain a subfamily of rank (exactly) four over ℚ(k) and determine a set of its free generators. Besides, we show there exist infinitely many elliptic curves of rank ≥ 5 parameterized by a rank five quartic elliptic curve. We also find a few particular examples with higher ranks. The families we construct have ℤ/2ℤ torsion subgroups in general. The previous work [13] has obtained similar Pythagorean quadruplet elliptic curve families in two parameters with rank ≥ 3. (Recall that by a Pythagorean quadruplet (a, b, c, d), we mean an integer solution to the quadratic equation a 2 + b 2 = c 2 + d 2.)

Root numbers of a family of elliptic curves and two applications

For each t∈Q∖{−1,0,1}, define an elliptic curve over Q by Et:y2=x(x+1)(x+t2).Using a formula for the root number W(Et) as a function of t and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves E/Q whose Mordell–Weil group contains Z×Z/2Z×Z/4Z, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.

Ranks of elliptic curves in cyclic sextic extensions of [formula omitted]

For an elliptic curve E/Q we show that there are infinitely many cyclic sextic extensions K/Q such that the Mordell–Weil group E(K) has rank greater than the subgroup of E(K) generated by all the E(F) for the proper subfields F⊂K. For certain curves E/Q we show that the number of such fields K of conductor less than X is ≫X.

Ranks of elliptic curves and deep neural networks

Ranks of elliptic curves and deep neural networks

Elliptic curve of rank at least 4 over $${\mathbb {Q}}(k)$$ with torsion point of order 4

Elliptic curve of rank at least 4 over $${\mathbb {Q}}(k)$$ with torsion point of order 4

Elliptic curve involving subfamilies of rank at least 5 over $\mathbb{Q}(t)$ or $\mathbb{Q}(t,k)$

Motivated by the work of Zargar and Zamani, we introduce a family of elliptic curves containing several one- (respectively two-) parameter subfamilies of high rank over the function field $\mathbb{Q}(t)$ (respectively $\mathbb{Q}(t,k)$). Following the approach of Moody, we construct two subfamilies of infinitely many elliptic curves of rank at least 5 over $\mathbb{Q}(t,k)$. Secondly, we deduce two other subfamilies of this family, induced by the edges of a rational cuboid containing five independent $\mathbb{Q}(t)$-rational points. Finally, we give a new subfamily induced by Diophantine triples with rank at least 5 over $\mathbb{Q}(t)$. By specialization, we obtain some specific examples of elliptic curves over $\mathbb{Q}$ with a high rank (8, 9, 10 and 11).

On the rank of elliptic curves arising from Pythagorean quadruplets, II

On the rank of elliptic curves arising from Pythagorean quadruplets, II

Integral triangles and perpendicular quadrilateral pairs with a common area and a common perimeter

By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $\geq 4$, and show the existence of infinitely many elliptic curves of rank $\geq 5$, parameterized by the points of an elliptic curve of positive rank.

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 the rank of elliptic curves arising from Pythagorean quadruplets

By a Pythagorean quadruplet $(a,b,c,d)$, we mean an integer solution to the quadratic equation $a^2 + b^2 = c^2 + d^2$. We use this notion to construct infinite families of elliptic curves of higher rank as far as possible. Furthermore, we give particular examples of rank eight.

Ranks of elliptic curves over $$\mathbb{Z}_p^2$$-extensions

Let E be an elliptic curve with good reduction at a fixed odd prime p and K an imaginary quadratic field where p splits. We give a growth estimate for the Mordell-Weil rank of E over finite extensions inside the $$\mathbb{Z}_p^2$$-extension of K.

Infinitely many elliptic curves of rank exactly two II

In this note, we construct an infinite family of elliptic curves $E$ defined over $\mathbf{Q}$ whose Mordell-Weil group $E(\mathbf{Q})$ has rank exactly two under the parity conjecture.

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$ y 2 = x 3 - m 2 x + p 2

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of $$E_{m,p} : y^2=x^3-m^2x+p^2$$ , where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of $$E_{m,p}(\mathbb {Q})$$ is trivial for both the cases { $$m\ge 1$$ , $$m\not \equiv 0\pmod 3$$ } and { $$m\ge 1$$ , $$m \equiv 0 \pmod 3$$ , with $$gcd(m,p)=1$$ }. We also show that given any odd prime p and for any positive integer m with $$m\not \equiv 0\pmod 3$$ and $$m\equiv 2\pmod {32}$$ , the lower bound for the rank of $$E_{m,p}(\mathbb {Q})$$ is 2. Finally, we find curves of rank 9 in this family.

On parametric spaces of bicentric quadrilaterals

Abstract In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by Ea : y 2 = x 3 + (a 4 − 4a 3 − 2a 2 − 4a + 1)x 2 + 16a 4 x which have, in general, (ℤ/8ℤ), and in rare cases (ℤ/2ℤ × ℤ/8ℤ) as torsion subgroups. We show the existence of infinitely many elliptic curves Ea of rank at least two with torsion subgroup ℤ/8ℤ, parametrised by the points of an elliptic curve of rank at least one, and give five particular examples of rank 5. We, also, show the existence of a subfamily of Ea whose torsion subgroup is ℤ/2ℤ × ℤ/8ℤ.

Mordell–Weil ranks of elliptic curves in the cyclotomic ℤ2-extension of the rationals

In this paper, we prove that there exist infinitely many elliptic curves whose Mordell–Weil ranks increase at the fourth layer of the cyclotomic [Formula: see text]-extension of [Formula: see text].

On the rank of elliptic curves with long arithmetic progressions

On the rank of elliptic curves with long arithmetic progressions

Elliptic curves of rank two and generalised Kato classes

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank \(>1\). This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted \(\kappa (f,g,h)\) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations \(V_g\) and \(V_h\) of \(\mathrm {Gal\,}(H/\mathbb {Q})\) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over \(\mathbb {Q}\) attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that \(\kappa (f,g,h)\) lies in the pro-p Selmer group of E over H precisely when \(L(E,V_{gh},1)=0\), where \(L(E,V_{gh},s)\) is the L-function of E twisted by \(V_{gh}:= V_g\otimes V_h\). In the setting of interest, parity considerations imply that \(L(E,V_{gh},s)\) vanishes to even order at \(s=1\), and the Selmer class \(\kappa (f,g,h)\) is expected to be trivial when \({\mathrm {ord}}_{s=1}L(E,V_{gh},s) >2\). The main new contribution of this article is a conjecture expressing \(\kappa (f,g,h)\) as a canonical point in \((E(H)\otimes V_{gh})^{G_\mathbb {Q}}\) when \({\mathrm {ord}}_{s=1} L(E,V_{gh},s)=2\). This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).

Infinitely many elliptic curves of rank exactly two

Infinitely many elliptic curves of rank exactly two

On the Rank of Elliptic Curves in Elementary Cubic Extensions

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED:y2+Dy=x3 (D∈Q∗) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L:Q] often becomes small.

Ranks of elliptic curves y^2=x^3\pm4px

Suppose that y 2 = x 3 +4px, y 2 = x 3 4px are elliptic curves where p is an odd prime. Then we treat ranks of these two curves according to the condition of prime number p. Mathematics Subject Classication: 11A41, 11G05