On rational points on conics

The Number of Rational Points on Conics Cp,k : x2 − ky2 = 1 over Finite Fields Fp

Let p be a prime number, Fp be a finite field, and let k ∈ Fp. In this paper, we consider the number of rational points on conics Cp,k : x − ky = 1 over Fp. We proved that the order of Cp,k over Fp is p− 1 if k is a quadratic residue mod p and is p+1 if k is not a quadratic residue mod p. Later we derive some results concerning the sums ∑ C [x] p,k(Fp) and ∑ C [y] p,k(Fp), the sum of x− and y−c...

Rational Points on Pencils of Conics and Quadrics with Many Degenerate Fibres

For any pencil of conics or higher-dimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics over Q, which is a consequence of recent advances in additive combinatorics.

Rational Points on Quartics

Rational points on curves

Rational points on curves

2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to `-adic representations 21 2.6 The isogen...