Amnon Besser

Finite and p-adic Polylogarithms

The finite nth polylogarithm lin(z) ∈ ℤ/p(z) is defined as ∑k=1p−1zk/kn. We state and prove the following theorem. Let Lik: ℂp → ℂp be the p-adic polylogarithms defined by Coleman. Then a certain linear combination Fn of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p1−nDFn(z) reduces modulo p>n+1 to lin−1(σ(z)), where D is the Cathelineau operator z(1−z)d/dz and σ is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.

Elliptic fibrations of K3 surfaces and QM Kummer surfaces

In this paper we show that the work of Nikulin on discriminant forms [12] can be used to construct elliptic fibrations of a K 3 surfaceX when the Néron-Severi lattice of X is known. By a K3 surface, in this work, we will always mean analgebraic K3 surface. As an application, we are able to write an equation describing a Kummer surface, Kummer (A), associated to a generic abelian surface A, whose ring of endomorphisms is isomorphic to a maximal order in a rational quaternion algebra of discriminants 6 or 15 (we call these Kummer surfaces QM Kummer surfaces hence the title of this work). By a lattice we mean a freeZ-moduleL, of finite rank, together with a nondegenerate symmetric bilinear form with values in Z, denotedx, y −→ x · y. With the exception of Sect. 4, we assume the lattice is even, i.e., that x · x ∈ 2Z for all x ∈ L. We letL(c) denote the lattice having the same underlyingZ-module asL but with the form multiplied byc. The signature of L will be denoted(+t+,−t−), wheret+ (resp.t−) is the number of +1’s (resp.−1’s) in the diagonalization of the underlying form over R. When S ⊂ T are two lattices, we say that T is anoverlatticeof S if S andT have the same rank. Our prime example here is the N éron-Severi lattice of complex surfaces and in particular of K3 and abelian surfaces. Let X be a compact complex surface. We let ρ(X) denote the Picard number of X. TheNéron-Severi lat-

Finite polynomial cohomology for general varieties

Nekovář and Nizioł (Syntomic cohomology and p-adic regulators for varieties over p-adic fields, 2013) have introduced in a version of syntomic cohomology valid for arbitrary varieties over p-adic fields. This uses a mapping cone construction similar to the rigid syntomic cohomology of (Besser, Israel J Math 120(1):291–334, 2000) in the good-reduction case, but with Hyodo–Kato (log-crystalline) cohomology in place of rigid cohomology. In this short note, we describe a cohomology theory which is a modification of the theory of Nekovář–Nizioł, modified by replacing

Computing integral points on hyperelliptic curves using quadratic Chabauty

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On an invariant related to a linear inequality

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A short proof of de Shalit’s cup product formula

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