An inverse Jacobian algorithm for Picard curves

We study the inverse Jacobian problem for the case of Picard curves over $${\mathbb {C}}$$ C . More precisely, we elaborate on an algorithm that, given a small period matrix $$\varOmega \in {\mathbb {C}}^{3\times 3}$$ Ω ∈ C 3 × 3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most $$4$$ 4 . In particular, we obtain the complete list of maximal CM Picard curves defined over $${\mathbb {Q}}$$ Q . In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

Evaluation of National Economy’s Development Balance: Structural and Dynamic Aspects

The work is devoted to the grounding of the scientific and methodological aspects and forming the practical recommendations for evaluation of national economy’s development balance. Assessment’s results are aimed at strict explanation of real economic dynamics to find relevant instruments and impact mechanisms to target national economic development to the mainstream. Structural and dynamic approach to the estimation gives an opportunity to neutralize the complexity of national economy as an object for the evaluation by means of building the phase portraits of national economic sectors’ movements in two-dimensional phase space structured with the balanced lines. Practicability of the “balanced growth concept” implementation as a part of the national economy’s balance development evaluating is scientifically proved. Using of the concept’s mechanisms and principles at the macro level of national economic systems’ evolution is proposed. The structural and logical schema of balanced indicators is formed. Empirical research of Ukrainian national economy’s balance development is done during 2008-2012’ period. Matrix of growth quality assessment is built for the main sectors of national economy. The conclusions about security, sustainability and expediency of national economic growth rates, its structural components are made. The national economy’s sectors characterized with the most sustainable and regular behavior according to assessment based on indicators of a Balanced Growth Rate and Sustainable Development Index are picked out. General results of balance and prospects of national economy’s development are formed.

Spaces of Abelian Differentials and Hitchin’s Spectral Covers

Using the embedding of the moduli space of generalized $GL(n)$ Hitchin’s spectral covers to the moduli space of meromorphic Abelian differentials we study the variational formulæ of the period matrix, the canonical bidifferential, the prime form and the Bergman tau function. This leads to residue formulæ which generalize the Donagi–Markman formula for variations of the period matrix. The computation of second derivatives of the period matrix reproduces the formula derived in [2] using the framework of topological recursion.

General Variational Formulas for Abelian Differentials

We use the jump problem technique developed in a recent paper [9] to compute the variational formula of any stable differential and its periods to arbitrary precision in plumbing coordinates. In particular, we give the explicit variational formula for the degeneration of the period matrix, easily reproving the results of Yamada [21] for nodal curves with one node and extending them to an arbitrary stable curve. Concrete examples are included. We also apply the same technique to give an alternative proof of the sufficiency part of the theorem in [1] on the closures of strata of differentials with prescribed multiplicities of zeroes and poles.

How to Identify an Arbitrage of Type B on Capital Markets

We propose here a 1-period matrix model of a fraction of the Polish financial market (for our purposes it will suffice to focus on a fraction of the market) built up from the point of view of the Polish biggest listed company KGHM. Using this model we construct an arbitrage portfolio consisting of 5 different assets, namely shares of KGHM, Treasury bills and 3 kinds of stock options. We recall the concept of arbitrage of type A and type B (called also an arbitrage I and arbitrage II, resp.) and illustrate it with examples. To prove that an arbitrage is possible to conduct, we separately distinguish scenarios when options prices are determined by the Black-Scholes formula, and when they deviate from their theoretical values. We prove that in all those cases an arbitrage of type B can be conducted. Since our approach does not rely on the specifics of Poland as a country, it can be equally well implemented in any other country which offers Treasury bills, as well as call and put options on shares of selected companies (KGHM in the studied case). The purpose of this study is to encourage practitioners to conduct an arbitrage in their own country, especially in a case when call and put options are offered on a local OTC market.

Constructing genus-3 hyperelliptic Jacobians with CM

Given a sextic CM field $K$, we give an explicit method for finding all genus-$3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$.

A resolution of the integration region problem for the supermoduli space integral

The integration region of the supermoduli space integral is defined in the super-Schottky group parametrization. The conditions on the super-period matrix elements are translated to relations on the parameters. An estimate of the superstring amplitude at arbitrary genus is sufficient for an evaluation of the cross-section to all orders in the expansion of the scattering matrix.