We formulate a detailed conjectural Eichler–Shimura type formula for the cohomology of local systems on Picard modular surface associated to group unitary similitudes $\operatorname{GU}(2,1, \mathbb{Q}(\sqrt{-3}))$. The is based counting points over finite fields curves genus three which are cyclic triple covers projective line. Assuming conjecture we able calculate traces Hecke operators space...

We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate problem in terms arithmetic field and its idèles, proceed to some extensions that arise naturally this context. then make use our computations determine fields moduli classify their Brauer groups. More specifically, provide an algorithm given a number K E, returns...

We show that for any N > 0 there exists a natural even n > N such that the discriminant of moduli of K3 surfaces of the degree n is not equal to the set of zeros of any automorphic form on the corresponding IV type domain. We give the necessary condition on a ”condition S ⊂ LK3 on Picard lattice of K3” for the corresponding moduli MS⊂LK3 of K3 to have the discriminant which is equal to the set ...

Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of K3 surface of Picard number zero, we shall clarify the structure of the automorphism group of a Picard generic hyperkähler manifold (Definition 1.4), in an optimal form up to finite group factor. Our argument uses Yau’s solution of Calabi’s conjecture, Dirichlet’s unit theorem and theory of Salem polynomials....

Infinite triangular matrices and, in particular, the redecoration operation on them, were studied by Matthes and Picard. In their work, redecoration is characterized as the cobind operation of what the authors call a “weak constructive comonad”. In this work, we identify weak constructive comonads as an instance of the more general notion of relative comonad. Afterwards, building upon the work ...

We compute the cohomology of the Picard bundle on the desingularization J̃ d (Y ) of the compactified Jacobian of an irreducible nodal curve Y . We use it to compute the cohomology classes of the Brill–Noether loci in J̃ d (Y ). We show that the moduli space M of morphisms of a fixed degree from Y to a projective space has a smooth compactification. As another application of the cohomology of the...

We continue our studies in higher order uniform convergence with rates and in Lp convergence with rates. Namely, in this article we establish some Lipschitz type results for the smooth Picard type singular integral operators and for the smooth GaussWeierstrass type singular integral operators. RESUMEN Continuamos nuestros estudios sobre convergencia uniforme de orden superior con radios y sobre...

Periodic solutions of certain large-scale systems of ODEs can be computed ee-ciently using a hybrid Newton{Picard scheme, especially in a continuation context. In this paper we describe and analyse how this approach can be extended to the direct computation of period doubling bifurcation points. The Newton{Picard scheme is based on shooting and a splitting of the state space in a low-dimensiona...

We study differential equations satisfied by modular forms of two variables associated to Γ1×Γ2, where Γi (i = 1, 2) are genus zero subgroups of SL2(R) commensurable with SL2(Z), e.g., Γ0(N) or Γ0(N)∗ for some N . In some examples, these differential equations are realized as the Picard–Fuchs differential equations of families of K3 surfaces with large Picard numbers, e.g., 19, 18, 17, 16. Our ...

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Néron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over th...