TY - JOUR

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

AU - Bhargava, Manjul

AU - Cremona, John

AU - Fisher, Tom

N1 - Funding Information:
We thank Benedict Gross, Marc Masdeu, Michael Stoll, Terence Tao, and Xiaoheng Wang for helpful conversations. The first author was supported by a Simons Investigator Grant and NSF Grant DMS-1001828, and thanks the Flatiron Institute for its kind hospitality during the academic year 2019–2020. The second author was supported by EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms, the Horizon 2020 European Research Infrastructures Project OpenDreamKit (#676541), and the Heilbronn Institute for Mathematical Research.
Publisher Copyright:
© 2021 The Author(s).

PY - 2021/5

Y1 - 2021/5

N2 - We consider the proportion of genus one curves over Q of the form z2 = f(x,y) where f(x,y) [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) [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) [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) [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

SN - 1793-0421

VL - 17

SP - 903

EP - 923

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 4

ER -