TY - JOUR

T1 - The proportion of genus one curves over Q defined by a binary quartic that everywhere locally have a point

AU - Bhargava, Manjul

AU - Cremona, John

AU - Fisher, Tom

N1 - Publisher Copyright:
© The Author(s).
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020

Y1 - 2020

N2 - We consider the proportion of genus one curves over Q of the form z2 = f(x, y) where f(x, y) ∈ Z[x, y] is a binary quartic form (or more generally of the form z2 + h(x, y)z = f(x, y) where also h(x, y) ∈ Z[x, y] is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime p is given by a fixed degree-9 rational function of p for all odd p (and for the generalized equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place.

AB - We consider the proportion of genus one curves over Q of the form z2 = f(x, y) where f(x, y) ∈ Z[x, y] is a binary quartic form (or more generally of the form z2 + h(x, y)z = f(x, y) where also h(x, y) ∈ Z[x, y] is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime p is given by a fixed degree-9 rational function of p for all odd p (and for the generalized equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place.

KW - Genus one curves

KW - Local solubililty

KW - Random diophantine equations

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U2 - 10.1142/S1793042121500147

DO - 10.1142/S1793042121500147

M3 - Article

AN - SCOPUS:85093506765

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

ER -