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We report on our project to find explicit examples of K3 surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are p-adic in nature.

In this paper we construct a natural category ~r of locally and topologically ringed spaces which contains both the category of locally noetherian formal schemes and the category of rigid analytic varieties as full subcategories. This category has applications in algebraic geometry and rigid analytic geometry. The idea of the definition of the category ~r is the following. From a formal point o...

In this paper we considered a moving from classical logic and Kolmogorov’s probability theory to non-classical p-adic valued logic and p-adic valued probability theory. Namely, we defined p-adic valued logic and further we constructed probability space for some ideals on truth values of p-adic valued logic. We proposed also p-adic valued inductive logic. Such a logic was considered for the firs...

On the p-adic Leopoldt transform of a power series BrunoAngì es 1 Let p be an odd prime number. Let X be the projective limit for the norm maps of the p-Sylow subgroups of the ideal class groups of Q(ζ p n+1), n ≥ 0. Let ∆ = Gal(Q(ζ p)/Q) and let θ be an even and non-trivial character of ∆. Then X is a Z p [[T ]]-module and the characteristic ideal of the isotypic component X(ωθ −1) is generate...

We define a p-adic character to be a continuous homomorphism from 1 + tFq [[t]] to Zp. For p > 2, we use the ring of big Witt vectors over Fq to exhibit a bijection between p-adic characters and sequences (ci)(i,p)=1 of elements in Zq , indexed by natural numbers relatively prime to p, and for which limi→∞ ci = 0. To such a p-adic character we associate an L-function, and we prove that this L-f...

The purpose of this paper is to study the arithmetic function f : Z+ → Q ∗ + defined by f(2l) = l (∀k, l ∈ N, l odd). We have, for example, f(1) = 1, f(2) = 1, f(3) = 3, f(12) = 1 3 , f(40) = 1 25 , . . . , so it is clear that f(n) is not always an integer. However, we will show in what follows that f satisfies the property that the product of the f(r) for 1 ≤ r ≤ n is always an integer, and it...

In the paper we develop the p-adic theory of discrete automata. Every automaton A (transducer) whose input/output alphabets consist of p symbols can be associated to a continuous (in fact, 1-Lipschitz) map from padic integers to p integers, the automaton function fA. The p-adic theory (in particular, the p-adic ergodic theory) turned out to be very efficient in a study of properties of automata...