Geometry-of-numbers methods are extended to count orbits in coregular spaces over arbitrary global fields, yielding bounds on average ranks and Selmer sizes for elliptic curves and hyperelliptic Jacobians.
Bhargava, Most hyperelliptic overQcurves have no rational points,http://arxiv.org/abs/ 1308.0395
2 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted projective space P(1,1,g+1) via an equation of the form C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the coefficients f_i lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all hyperelliptic curves over Q of genus g by height. Then we prove that, as g tends to infinity: 1) a density approaching 100% of hyperelliptic curves of genus g have no rational points; 2) a density approaching 100% of those hyperelliptic curves of genus g that have points everywhere locally fail the Hasse principle; and 3) a density approaching 100% of hyperelliptic curves of genus g have empty Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point. We also prove positive proportion results of this type for individual genera, including g = 1.
fields
math.NT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Proves insolvability of ax^d - c(m²y² + n²z²) + xyz - b = 0 for b = a(2cmn)^d - l c^s t^{2q} with l prime ≡3 mod 4 (and odd powers), except certain x mod 4l, plus bounds and hyperelliptic torsion results.
citing papers explorer
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Geometry-of-numbers methods over global fields II: Coregular representations
Geometry-of-numbers methods are extended to count orbits in coregular spaces over arbitrary global fields, yielding bounds on average ranks and Selmer sizes for elliptic curves and hyperelliptic Jacobians.
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Weighted Fruit Diophantine Equations and Hyperelliptic Curves
Proves insolvability of ax^d - c(m²y² + n²z²) + xyz - b = 0 for b = a(2cmn)^d - l c^s t^{2q} with l prime ≡3 mod 4 (and odd powers), except certain x mod 4l, plus bounds and hyperelliptic torsion results.