On the average value of the canonical height in higher dimensional families of elliptic curves

Wei Pin Wong

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Displaying similar documents to “On the average value of the canonical height in higher dimensional families of elliptic curves”

Some examples of 5 and 7 descent for elliptic curves over $Q$

Tom Fisher (2001)

Journal of the European Mathematical Society

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We perform descent calculations for the families of elliptic curves over $Q$ with a rational point of order $n = 5$ or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over $Q$ may become arbitrarily large.

On invariants of elliptic curves on average

Amir Akbary, Adam Tyler Felix (2015)

Acta Arithmetica

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We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_{E} (p)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field $_{p}$ . Let be the family of elliptic curves $E_{a, b} : y^{2} = x^{3} + a x + b$ , where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1 / | | \sum_{E \in} \sum_{p \leq x} e_{E}^{k} (p) = C_{k} l i (x^{k + 1}) + O ((x^{k + 1}) / {(l o g x)}^{c}$ )as x → ∞, as long...

Small discriminants of complex multiplication fields of elliptic curves over finite fields

Igor E. Shparlinski (2015)

Czechoslovak Mathematical Journal

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We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field $𝔽_{p}$ of $p$ elements, such that the discriminant $D (E)$ of the quadratic number field containing the endomorphism ring of $E$ over $𝔽_{p}$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

Elliptic curves with $ℚ (ℰ [3]) = ℚ (ζ_{3})$ and counterexamples to local-global divisibility by 9

Laura Paladino (2010)

Journal de Théorie des Nombres de Bordeaux

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We give a family $ℱ_{h, β}$ of elliptic curves, depending on two nonzero rational parameters $β$ and $h$ , such that the following statement holds: let $ℰ$ be an elliptic curve and let $ℰ [3]$ be its 3-torsion subgroup. This group verifies $ℚ (ℰ [3]) = ℚ (ζ_{3})$ if and only if $ℰ$ belongs to $ℱ_{h, β}$ . Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer...

The Mordell-Weil bases for the elliptic curve $y^{2} = x^{3} - m^{2} x + m^{2}$

Sudhansu Sekhar Rout, Abhishek Juyal (2021)

Czechoslovak Mathematical Journal

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Let $D_{m}$ be an elliptic curve over $ℚ$ of the form $y^{2} = x^{3} - m^{2} x + m^{2}$ , where $m$ is an integer. In this paper we prove that the two points $P_{- 1} = (- m, m)$ and $P_{0} = (0, m)$ on $D_{m}$ can be extended to a basis for $D_{m} (ℚ)$ under certain conditions described explicitly.

A local-global principle for rational isogenies of prime degree

Andrew V. Sutherland (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a number field. We consider a local-global principle for elliptic curves $E / K$ that admit (or do not admit) a rational isogeny of prime degree $ℓ$ . For suitable $K$ (including $K = ℚ$ ), we prove that this principle holds for all $ℓ \equiv 1 mod 4$ , and for $ℓ < 7$ , but find a counterexample when $ℓ = 7$ for an elliptic curve with $j$ -invariant $2268945 / 128$ . For $K = ℚ$ we show that, up to isomorphism, this is the only counterexample.

On annealed elliptic Green's function estimates

Daniel Marahrens, Felix Otto (2015)

Mathematica Bohemica

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We consider a random, uniformly elliptic coefficient field $a$ on the lattice $ℤ^{d}$ . The distribution $〈 \cdot 〉$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G (t, x, y)$ satisfy optimal annealed estimates which are $L^{2}$ and $L^{1}$ , respectively, in probability, i.e., they obtained bounds on $〈 | \nabla_{x} G (t, x, y) {|^{2} 〉}^{1 / 2}$ and $〈 | \nabla_{x} \nabla_{y} G (t, x, y) | 〉$ . In particular, the elliptic Green’s function $G (x, y)$ satisfies optimal annealed bounds. In their recent work,...

Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah, Anup Das, Azizul Hoque (2024)

Archivum Mathematicum

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We consider the Lebesgue-Ramanujan-Nagell type equation $x^{2} + 5^{a} 13^{b} 17^{c} = 2^{m} y^{n}$ , where $a, b, c, m \geq 0, n \geq 3$ and $x, y \geq 1$ are unknown integers with $gcd (x, y) = 1$ . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$ -integral points on a class of elliptic curves.