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Yaacov Kopeliovich

Publications and source records attributed to Yaacov Kopeliovich.

At least 19 recordsLinked to original sources
arXiv2026

Schwarz maps for modular curves

We solve a classical problem posed by F. Klein and studied by A. Hurwitz concerning the construction of linear ordinary differential equations associated with modular transformations of fixed degree. For every odd integer $N\ge 3$ (respectively, even integer $N\ge 4$), we construct a canonical invariant model of the modular curve $X(N)=\mathbb{H}/Γ(N)$ (respectively, $X_H(N)=\mathbb{H}/H(N)$ where $H(N)=Γ(N)\capΓ_0^0(2N)$), together with a linear ordinary differential equation with rational coefficients whose Schwarz map parametrizes this model and whose projective monodromy group is the finite quotient $PSL_2(\mathbb{Z})/\tildeΓ(N)$ (respectively, $PSL_2(\mathbb{Z})/\tilde{H}(N)$). The construction is expressed in terms of invariant projective geometry and Picard-Vessiot theory and yields equations that are canonical up to projective equivalence. In this framework, Hurwitz's classical equation for degree $7$ appears as a special case of a general mechanism. The results place Klein's question within the modern theory of algebraic linear ordinary differential equations and provide a uniform geometric realization of modular transformation groups as projective differential Galois groups. As an application, we construct an explicit example of a linear ordinary differential equation associated with $X(9)$.

math.NT↗
arXiv2025

The fundamental group of compact Riemann surface

We derive presentation and relations for a group of compact Riemann surface that is given as branched cover of the sphere. In the case that one of the permutations is of full cycle of the form $(1...n)$ we derive a straightforward process to obtain the standard presentation of the fundamental group of Algebraic curve in the form $\prod_{i=1}^g[a_i,b_i]=1$

math.CV↗
arXiv2025

The Fundamental Group of a Compact Riemann Surface via Branched Covers

Let $X$ be a compact Riemann surface of genus $g$ and let $x \in X$. We derive the classical presentation of $π_1(X,x)$ (i.e the one given by $2g$ generators $a_1,b_1, \dots, a_g,b_g$ and the relation $\prod_{i=1}^g[a_i,b_i] = 1$) from the description of $X$ as a branched cover $f : X \to \mathbb{C}\mathbb{P}^1$.

math.AT↗
arXiv2025

Algebraic solution of the Jacobi inverse problem and explicit addition laws

We formulate a solution to the Algebraic version of the Inverse Jacobi problem. Using this solution we produce explicit addition laws on any algebraic curve generalizing the law suggested by Leykin [2] in the case of (n, s) curves. This gives a positive answer to a question asked by T. Shaska whether addition laws appearing in [2] can be produced in a coordinate free manner.

math.CV↗
arXiv2024

Portfolio Optimization with Feedback Strategies Based on Artificial Neural Networks

With the recent advancements in machine learning (ML), artificial neural networks (ANN) are starting to play an increasingly important role in quantitative finance. Dynamic portfolio optimization is among many problems that have significantly benefited from a wider adoption of deep learning (DL). While most existing research has primarily focused on how DL can alleviate the curse of dimensionality when solving the Hamilton-Jacobi-Bellman (HJB) equation, some very recent developments propose to forego derivation and solution of HJB in favor of empirical utility maximization over dynamic allocation strategies expressed through ANN. In addition to being simple and transparent, this approach is universally applicable, as it is essentially agnostic about market dynamics. To showcase the method, we apply it to optimal portfolio allocation between a cash account and the S&P 500 index modeled using geometric Brownian motion or the Heston model. In both cases, the results are demonstrated to be on par with those under the theoretical optimal weights assuming isoelastic utility and real-time rebalancing. A set of R codes for a broad class of stochastic volatility models are provided as a supplement.

q-fin.PM↗
arXiv2024

On Merton's Optimal Portfolio Problem with Sporadic Bankruptcy for Isoelastic Utility

We consider a stock that follows a geometric Brownian motion (GBM) and a riskless asset continuously compounded at a constant rate. We assume that the stock can go bankrupt, i.e., lose all of its value, at some exogenous random time (independent of the stock price) modeled as the first arrival time of a homogeneous Poisson process. For this setup, we study Merton's optimal portfolio problem consisting in maximizing the expected isoelastic utility of the total wealth at a given finite maturity time. We obtain an analytical solution using coupled Hamilton-Jacobi-Bellman (HJB) equations. The optimal strategy bans borrowing and never allocates more wealth into the stock than the classical Merton ratio recommends. For non-logarithmic isoelastic utilities, the optimal weights are non-myopic. This is an example where a realistic problem, being merely a slight modification of the usual GBM model, leads to non-myopic weights. For logarithmic utility, we additionally present an alternative derivation using a stochastic integral and verify that the weights obtained are identical to our first approach. We also present an example for our strategy applied to a stock with non-zero bankruptcy probability.

q-fin.MF↗
arXiv2022

Addition via reduction algorithm on trigonal curves

In this paper we propose a direct and explicit realization of addition of divisors by means of an iterative reduction algorithm. Each iteration of the algorithm is the reduction of a degree $g+1$ divisor to a divisor of degree~$g$. Such an approach allows to carry out all computations explicitly in a symbolic form, which is done for curves $C_{3,4}$, $C_{3,5}$ in this paper, and also for curves of higher genera up to $C_{3,13}$, $C_{3,14}$.

math.AG↗
arXiv2022

Solution of Mumford's second problem

A complete solution of Mumford's second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. Expressions for theta derivatives appear to be homogeneous of degree $3$ with respect to theta constants.

math.CV↗
arXiv2020

Thomae's Derivative Formulae for Trigonal Curves

In this paper we prove a Thomae derivative formula for trigonal curves admitting a non-singular affine model. This formula relates the derivatives of theta functions with rational characteristics on the curve to explicit expressions in the branching values.

math.AG↗
arXiv2020

Thomae formula for Abelian covers of $\mathbb{CP}^{1}$

Abelian covers of $\mathbb{CP}^{1}$, with fixed Galois group $A$, are classified, as a first step, by a discrete set of parameters. Any such cover $X$, of genus $g\geq1$ say, carries a finite set of $A$-invariant divisors of degree $g-1$ on $X$ that produce non-zero theta constants on $X$. We show how to define a quotient involving a power of the theta constant on $X$ that is associated with such a divisor $Δ$, some polynomial in the branching values, and a fixed determinant on $X$ that does not depend on $Δ$, such that the quotient is constant on the moduli space of $A$-covers with the given discrete parameters. This generalizes the classical formula of Thomae, as well as all of its known extensions by various authors.

math.AG↗
arXiv2020

On Spaces Associated with Invariant Divisors on Galois Covers of Riemann Surfaces and Their Applications

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.

math.AG↗
arXiv2020

Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of solution of the Jacobi inversion problem which provides the way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details comparing with the known in cryptography, and is extended to divisors of arbitrary degrees comparing with the known in the theory of hyperelliptic functions.

math.AG↗
arXiv2019

The addition on Jacobian varieties from a geometric viewpoint

We give a geometric interpretation of the group law for Jacobian varieties by extending the geometric construction of chords and tangents on an elliptic curve. For any given algebraic curve $\mathcal X$ and reduced divisors $D_1, D_2 \in \mbox{Jac } \mathcal X$, we define curves $\mathcal X^\prime$ and $\mathcal X^{"}$ such that the intersection $\mathcal X \cap \mathcal X^\prime$ determines precisely the divisor $-(D_1+D_2)$ and the intersection $\mathcal X\cap \mathcal X^{"}$ determines $D_1+D_2$. For superelliptic curves such formulas are made explicit.

math.NT↗
arXiv2016

Thomae formula for $2$ Abelian covers of $\mathbb{CP}^1$

Let $X$ be an Abelian cover $\mathbb{CP}^{1}$ ramified at $mr$ points, $λ_1...λ_{mr}.$ we define a class of non positive divisors on $X$ of degree $g-1$ supported in the pre images of the branch points on $X$, such that the Riemann theta function doesn't vanish on their image in $J(X).$ We obtain a Thomae formula similar to the formulas [BR],[Na],[Z] ,[EG] and [Ko]. We show that up to a certain determinant of the non standard periods of $X$, the value of the Riemann theta function at these divisors raised to a high enough power is a polynomial in the branch point of the curve $X.$ Our approach is based on a refinement of Accola's results and Nakayashiki's approach explained in [Na] for Abelian covers.

math.CV↗
arXiv2015

General cyclic covers and their Thomae formula

Let $X$ be a general cyclic cover of $\mathbb{CP}^{1}$ ramified at $m$ points, $λ_1...λ_m.$ we define a class of non positive divisors on $X$ of degree $g-1$ supported in the pre images of the branch points on $X$, such that the the standard theta function doesn't vanish on their image in $J(X).$ These divisors generalize the divisors introduced in [BR] and [Na]. Generalizing the results of [BR],[Na] and [EG] we show that up to a certain determinant of the non standard periods of $X$, the value of the theta functions at these divisors is a polynomial in the branch point of the curve $X.$ Our treatment is based on a generalization of Accola's results of the 3 cyclic sheeted cover [Ac1] and a straightforward generalization of Nakayashiki's approach explained in [Na] in the non singular case for any singular cyclic cover.

math.CV↗
arXiv2009

On theta functions of order four

We prove that the fourth powers of theta functions with even characteristics form a basis of the space of even theta functions of order four on a principally polarized Abelian variety without vanishing theta-null.

math.AG↗