Local and global densities for Weierstrass models of elliptic curves

Cremona, John E.; Sadek, Mohammad

doi:10.4310/mrl.2023.v30.n2.a5

Cited by 5 publications

(1 citation statement)

References 26 publications

(107 reference statements)

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“…Proof It is known that, for any integer

D

, when ordered by the height of the coefficients, there is a positive proportion of elliptic curves over

\mathbb {Q}

whose conductor is coprime to

D

(see, e.g. Theorem 4.2(2) of [11]; their ordering is slightly different, which may result in a different density, but we only need to know that this density is non‐zero). In other words,

\lim _{X\rightarrow \infty }\frac{\#N}{\#T}=k&gt;0

where

T

is the set of curves

E_{A,B}

such that

H(E_{A,B})&lt;X

and

N\subset T

is the set of elliptic curves

E_{A,B}

such that

(N_E,D)=1

.…”

Section: Minimalist Conjecture For Twistsmentioning

confidence: 99%

Root numbers and parity phenomena

Kellock,

Dokchitser

2023

Bulletin of London Math Soc

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The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton‐Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at a ‘user’ and, as such, we will not be concerned with the proofs of known cases of the parity conjecture, but instead, we will demonstrate the use of the theory by means of examples.

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“…Proof It is known that, for any integer

D

, when ordered by the height of the coefficients, there is a positive proportion of elliptic curves over

\mathbb {Q}

whose conductor is coprime to

D

(see, e.g. Theorem 4.2(2) of [11]; their ordering is slightly different, which may result in a different density, but we only need to know that this density is non‐zero). In other words,

\lim _{X\rightarrow \infty }\frac{\#N}{\#T}=k&gt;0

where

T

is the set of curves

E_{A,B}

such that

H(E_{A,B})&lt;X

and

N\subset T

is the set of elliptic curves

E_{A,B}

such that

(N_E,D)=1

.…”

Section: Minimalist Conjecture For Twistsmentioning

confidence: 99%

Root numbers and parity phenomena

Kellock,

Dokchitser

2023

Bulletin of London Math Soc

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On the distribution of analytic ranks of elliptic curves

Cho,

Jeong

2023

Math. Z.

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Densities on Dedekind domains, completions and Haar measure

Demangos,

Longhi

2024

Math. Z.

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Let D be the ring of S-integers in a global field and D its profinite completion. Given X ⊆ D n , we consider its closure X ⊆ D n and ask what can be learned from X about the "size" of X .In particular, we ask when the density of X is equal to the Haar measure of X . We provide a general definition of density which encompasses the most commonly used ones. Using it we give a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. We also show how Ekedahl's sieve fits into our setting and find conditions ensuring that X can be written as a product of local closures. In another direction, we extend the Davenport-Erdős theorem to every D as above and offer a new interpretation of it as a "density=measure" result. Our point of view also provides a simple proof that in any D the set of elements divisible by at most k distinct primes has density 0 for any k ∈ N. Finally, we show that the closure of the set of prime elements of D is the union of the group of units of D with a negligible part.

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Local and global densities for Weierstrass models of elliptic curves

Cited by 5 publications

References 26 publications

Root numbers and parity phenomena

Root numbers and parity phenomena

On the distribution of analytic ranks of elliptic curves

Densities on Dedekind domains, completions and Haar measure

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