The defect of weak approximation for homogeneous spaces
The density of integral points on hypersurfaces of degree at least four
The density of rational points on a pfaff curve
This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.
The density of rational points on cubic surfaces
The dimension of some affine Deligne–Lusztig varieties
The Diophantine equation f(x) = g(y)
The Diophantine Equation X³ = u+v over Real Quadratic Fields
Let k be a real quadratic field and let and be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v (, ) is developed.
The diophantine equation X3 + Y3 = DZ3 and the conjectures of Birch and Swinnerton-Dyer.
The direct image theorem in formal and rigid geometry.
The distribution of group structures on elliptic curves over finite prime fields.
The Drinfeld Modular Jacobian has connected fibers
We study the integral model of the Drinfeld modular curve for a prime . A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod . A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of which, after contractions in...
The equation x+y = α in multiplicatively dependent unknowns
The equation xyz=x+y+z=1 in integers of a quartic field
The Erdős and Halberstam theorems for Drinfeld modules of any rank (with an appendix by Hugh Thomas)
The existence of infinitely many supersingular primes for every elliptic curve over Q.
The field of moduli of abelian surfaces with complex multiplication.
The field of moduli of quaternionic multiplication on abelian varieties.
The fluctuations in the number of points on a family of curves over a finite field
Let be a positive integer, a finite field of cardinality with . In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of -points on the curve given by the affine model , where is drawn at random uniformly from the set of all monic -th power-free polynomials of degree as . The method also enables us to study the fluctuations in the number of -points on the same family of curves arising from the set of monic irreducible...
The formal completion of the Néron model of J0(p).
For any prime number p > 3 we compute the formal completion of the Néron model of J0(p) in terms of the action of the Hecke algebra on the Z-module of all cusp forms (of weight 2 with respect to Γ0(p)) with integral Fourier development at infinity.
The genus of curves over finite fields with many rational points.