One attempt to the modular function I
One attempt to the modular function - II
One-dimensional Hurwitz spaces, modular curves, and real forms of Belyi meromorphic functions.
One-Dimensional Local Rings with Reduced Associated Graded Ring and Their Hilbert Functions.
Open surfaces with non-positive Euler characteristic
Operads and algebraic geometry.
Operateurs differentiels de Shimura et espaces prehomogenes.
Opérateurs différentiels et courbes elliptiques
Opérateurs différentiels sur les espaces analytiques.
Opération de Cartier et vecteurs de Witt
Operations of Points on Elliptic Curve in Projective Coordinates
In this article, we formalize operations of points on an elliptic curve over GF(p). Elliptic curve cryptography [7], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: compellProjCo and addellProjCo are unary and binary operations of a point over the elliptic curve.
Optimal bound for the number of -curves on extremal rational surfaces.
Optimal curves differing by a 3-isogeny
Stein and Watkins conjectured that for a certain family of elliptic curves E, the X₀(N)-optimal curve and the X₁(N)-optimal curve of the isogeny class 𝓒 containing E of conductor N differ by a 3-isogeny. In this paper, we show that this conjecture is true.
Optimal curves differing by a 5-isogeny
For i = 0,1, let be the -optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational point of order...
Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.
We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model T to be realized has only double singularities and gives an algebraic curve of degree 2N+2K, where N and K are the numbers of double points and connected components of T. This degree is optimal in the sense that for any choice of the numbers N and K there exist models which cannot be realized algebraically with...
Optimal destabilizing vectors in some Gauge theoretical moduli problems
We prove that the well-known Harder-Narsimhan filtration theory for bundles over a complex curve and the theory of optimal destabilizing -parameter subgroups are the same thing when considered in the gauge theoretical framework.Indeed, the classical concepts of the GIT theory are still effective in this context and the Harder-Narasimhan filtration can be viewed as a limit object for the action of the gauge group, in the direction of an optimal destabilizing vector. This vector appears as an extremal...
Optimal levels for modular mod 2 representations over totally real fields.
Optimality of the Width- Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases
We consider digit expansions with an endomorphism of an Abelian group. In such a numeral system, the -NAF condition (each block of consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight admits an optimal -NAF).This result is then applied to imaginary quadratic bases, which are used for scalar multiplication in elliptic...
Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface (II).
In [6], orbifold G-bundles on a certain class of elliptic fibrations over a smooth complex projective curve X were related to parabolic G-bundles over X. In this continuation of [6] we define and investigate holomorphic connections on an orbifold G-bundle over an elliptic fibration.
Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface.
Let X be a compact Riemann surface and associated to each point p-i of a finite subset S of X is a positive integer m-i. Fix an elliptic curve C. To this data we associate a smooth elliptic surface Z fibered over X. The group C acts on Z with X as the quotient. It is shown that the space of all vector bundles over Z equipped with a lift of the action of C is in bijective correspondence with the space of all parabolic bundles over X with parabolic structure over S and the parabolic weights at any...