Random Weierstrass equations over Z_p have Haar-measure densities for admitting a Q_p-rational 3-torsion point or 3-isogeny given by exact rational functions depending on p modulo 3.
Local solubility of a family of ternary conics over a biprojective base I
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Let $f,g\in\mathbb{Z}[u_1,u_2]$ be binary quadratic forms. We provide upper bounds for the number of rational points $(u,v)\in\mathbb{P}^1(\mathbb{Q})\times\mathbb{P}^1(\mathbb{Q})$ such that the ternary conic \[ X_{(u,v)}: f(u_1,u_2)x^2 + g(v_1,v_2)y^2 = z^2 \] has a rational point. We also give some conditions under which lower bounds exist.
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The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny
Random Weierstrass equations over Z_p have Haar-measure densities for admitting a Q_p-rational 3-torsion point or 3-isogeny given by exact rational functions depending on p modulo 3.