Arithmetic geometry

Rational points of varieties over number fields: torsion and rank of elliptic curves, naive heights, Hasse principle, local-global failures.

• Hasse principle for random Fano hypersurfaces
• Heights of rational points on elliptic curves
• Local-global conjecture for Apollonian circle packings
• Mazur's torsion theorem
• Ranks of the elliptic curves y² = x³ − 2pqx

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