The paper investigates invariants of compactified Picard modular surfaces by principal congruence subgroups of Picard modular groups. The applications to the surface classification and modular forms are also discussed.

In general, Picard-Fuchs systems in N = 2 supersymmetric Yang-Mills theories are realized as a set of simultaneous partial differential equations. However, if the QCD scale parameter is used as unique independent variable instead of moduli, the resulting Picard-Fuchs systems are represented by a single ordinary differential equation (ODE) whose order coincides with the total number of independe...

There is an Mayer-Vietoris exact sequence 0 → Vn → PicZCpn → Cl Q(ζn−1) × Pic ZCpn−1 → 0 involving the Picard group of the integer group ring ZCpn where Cpn is the cyclic group of order p and ζn−1 is a primitive p -th root of unity. The group Vn splits as Vn ∼= V + n ⊕ V − n and V − n is explicitly known. V + n is a quotient of an in some sense simpler group Vn. In 1977 Kervaire and Murthy conj...

In this article we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.

We describe all of the smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5.

The Pleistocene vertebrate assemblage of Aldabra Atoll has been comparatively well studied. Three Upper Pleistocene fossil localities have been described yielding birds, reptiles and terrestrial molluscs. Those of Bassin Cabri and Bassin Lebine on Ile Picard are undated but must be in excess of 136,000 YBP, whereas Point Hodoul on Malabar Island is circa 100,000 YBP. Aldabra was seemingly compl...

Linear ordinary differential equations (ODEs) with periodic coefficients appear in various interesting applications, such as determining the linear stability regions of systems of vertically driven multiple pendula. Sinha and Butcher [1, 2] have obtained very good approximations to the solutions of such equations by calculating approximate Picard iterates symbolically in the parameters on which...

Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implement...

In this paper we construct the first known explicit family of K3 surfaces defined over the rationals that are proved to have geometric Picard number 3. This family is dense in one of the components of the moduli space of all polarized K3 surfaces with Picard number at least 3. We also use an example from this family to fill a gap in an earlier paper by the first author. In that paper, an argume...

We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of min...