For a hyperelliptic curve of genus g, it is well known that the symmetric products of g points on the curve are expressed in terms of their Abel–Jacobi image by the hyperelliptic sigma function (Jacobi inversion formulae). Matsutani and Previato gave a natural generalization of the formulae to the more general algebraic curves defined by y = f(x), which are special cases of (n, s) curves, and d...

by the mordell-weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎there is no known algorithm for finding the rank of this group‎. ‎this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1 +X)(1 + Y )(X + Y )− αXY , α ∈ R.

we consider the number of zeros of the integral $i(h) = oint_{gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. we prove that the number of zeros of $i(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

Derivation of standard curves for the EMIT therapeutic drug monitoring system involves several mathematical algorithms, all of which can be rewritten in the form of a linear equation y = mx + b. We examined the stability of the standard curve in terms of slope and y-intercept for three drug assays (procainamide, gentamicin, and carbamazepine) by generating calibration curves intermittently for ...

Let p be a prime number, Fp be a finite field and t ∈ Fp = Fp − {0}. In this paper we obtain some properties of elliptic curves Ep,t : y = y = x − tx over Fp. In the first section we give some notations and preliminaries from elliptic curves. In the second section we consider the rational points (x, y) on Ep,t. We give a formula for the number of rational points on Ep,t over Fp for an integer n...

We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C1. In general, we show that a large (but in general proper) subgroup of the 2torsion classes are given by the construction. Examples ...

In this brief note, we prove the following two Artin-Schreier curves are supersin-gular.

Among known algebro-geometric constructions that have the flavor of the space of unparametrized paths, one should mention the Kontsevich moduli stack M g,2(X, β) of stable maps from 2-pointed curves (C, x, y) of genus g into X, see [Ma]. Indeed, this stack consists of maps (C, x, y) → X for variable (C, x, y) modulo isomorphisms of curves, i.e., changes of parametrization. The points x and y se...

In this paper, we study Riccati solutions of Painlevé equations from a view point of geometry of Okamoto-Painlevé pairs (S,Y ). After establishing the correspondence between (rational) nodal curves on S − Y and Riccati solutions, we give the complete classification of the configurations of nodal curves on S − Y for each Okamoto–Painlevé pair (S, Y ). As an application of the classification, we ...