Corrections to the paper "On values of a polynomial a arithmetic progressions with equal products" (Acta Arith. 72 (1995), 67-76)
Corrigendum to "Representatons of multivariate polynomials by sums of univariate polynomials in linear forms" (Colloq. Math. 112 (2008), 201-233)
Corrigendum to the paper "Diophantine equations in cyclotomic fields".
Corrigendum to the paper "The number of solutions of the Mordell equation" (Acta Arith. 88 (1999), 173–179)
Counting all equilateral triangles in .
Counting rational points on cubic and quartic surfaces
Counting rational points on del Pezzo surfaces with a conic bundle structure
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
Counting solutions of decomposable form equations
Counting with Gauss' sums.
Courbes modulaires et courbes de Fermat.
Criterion for 3 to be eleventh power
Crossed ladders and Euler's quartic.
Cubic forms, powers of primes and the Kraus method
We consider the Diophantine equation , where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).
Cubic moments of Fourier coefficients and pairs of diagonal quartic forms
We establish the non-singular Hasse principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.