The equation xyz = x + y + z = 1 in integers of a cubic field.
The existence of infinitely many supersingular primes for every elliptic curve over Q.
The field of definition of the Mordell-Weil group of an elliptic curve over a function field
The field of moduli of quaternionic multiplication on abelian varieties.
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 fundamental group of a Galois cover of .
The genus of curves over finite fields with many rational points.
The global structure of moduli spaces of polarized -divisible groups.
The group of automorphisms of algebraic function fields.
The groups of points on abelian varieties over finite fields
Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).
The Hasse principle for homogeneous spaces.
The Hasse problem for rational surfaces.
The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes.
The Hilbert Modular Surface for the Ideal (...5) and the Horrocks-Mumford Bundle.
The Hirzebruch-Mumford volume for the orthogonal group and applications.
The indecomposability of H*(G/B, L)
The integral points on elliptic curves
Let be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if and both and are odd primes, then the general elliptic curve has only the integral point . By this result we can get that the above elliptic curve has only the trivial integral point for etc. Thus it can be seen that the elliptic curve really is an unusual elliptic curve which has large integral points.
The intrinsic Hodge theory of -adic hyperbolic curves.
The Lefschetz trace formula for algebraic stacks.
The local lifting problem for actions of finite groups on curves
Let be an algebraically closed field of characteristic . We study obstructions to lifting to characteristic the faithful continuous action of a finite group on . To each such a theorem of Katz and Gabber associates an action of on a smooth projective curve over . We say that the KGB obstruction of vanishes if acts on a smooth projective curve in characteristic in such a way that and have the same genus for all subgroups . We determine for which the KGB obstruction...