A basis for the non-archimedean holomorphic theta functions.
A Bogomolov property for curves modulo algebraic subgroups
Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
A class of diophantine equations connected with certain elliptic curves over
A Classical Diophantine Problem and Modular Forms of Weight 3/2.
A computer algebra solution to a problem in finite groups.
We report on a partial solution of the conjecture that the class of finite solvable groups can be characterised by 2-variable identities. The proof requires pieces from number theory, algebraic geometry, singularity theory and computer algebra. The computations were carried out using the computer algebra system SINGULAR.
A construction of curves over finite fields
A counterexample in the theory of local zeta functions.
A criterion for rational places over local fields.
A descent map for curves with totally degenerate semi-stable reduction
Let be a local field of residue characteristic . Let be a curve over whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to- rational torsion subgroup on the Jacobian of . We also determine divisibility of line bundles on , including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of .
A dimension formula for Ekedahl-Oort strata
We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by the classes in a Weyl group modulo a subgroup, and each class has a distinguished representative of minimal length. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.
A Diophantine problem on algebraic curves over function fields of positive characteristic
A diophantine system.
A dynamical Shafarevich theorem for twists of rational morphisms
Let K denote a number field, S a finite set of places of K, and ϕ: ℙⁿ → ℙⁿ a rational morphism defined over K. The main result of this paper states that there are only finitely many twists of ϕ defined over K which have good reduction at all places outside S. This answers a question of Silverman in the affirmative.
A family of Kummer covers over the Hermitian function field.
A family of varieties with exactly one pointless rational fiber
We construct a concrete example of a -parameter family of smooth projective geometrically integral varieties over an open subscheme of such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.
A finiteness result for the compactly supported cohomology of rigid analytic varieties, II
Let be a separated morphism of adic spaces of finite type over a non-archimedean field with affinoid and of dimension , let be a locally closed constructible subset of and let be the morphism of pseudo-adic spaces induced by . Let be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of and let be a constant -module of finite type on . There is a natural class of -modules on generated by the constructible -modules...
A finiteness theorem for the symmetric square of an elliptic curve.
A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction.
A fixed point formula for varieties over finite fields.
A fixed point formula of Lefschetz type in Arakelov geometry II: A residue formula
This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "à la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut- Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.