Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Interpolation for Restricted Tangent Bundles of General Curves

Abstract: Let (C, p_1, p_2, \ldots, p_n) be a general marked curve of genus g, and q_1, q_2, ..., q_n \in P^r be a general collection of points. We determine when there exists a nondegenerate degree d map f : C \to P^r so that f(p_i) = q_i for all i. This is a consequence of our main theorem, which states that the restricted tangent bundle f^* T_{P^r} of a general curve of genus g, equipped with a general degree d map f to P^r, satisfies the property of interpolation (i.e.\ that for a general effective divisor D of any degree on C, either H^0(f^* T_{P^r}(-D)) = 0 or H^1(f^* T_{P^r}(-D)) = 0). We also prove an analogous theorem for the twist f^* T_{P^r}(-1).

The Riemann constant for a non-symmetric Weierstrass semigroup

Abstract: The zero divisor of the theta function of a compact Riemann surface X of genus g is the canonical theta divisor of Pic\({^{(g-1)}}\) up to translation by the Riemann constant \({\Delta}\) for a base point P of X. The complement of the Weierstrass gaps at the base point P gives a numerical semigroup, called the Weierstrass semigroup. It is classically known that the Riemann constant \({\Delta}\) is a half period, namely an element of \({\frac{1}{2} \Gamma_\tau}\) , for the Jacobi variety \({\mathcal{J}(X)=\mathbb{C}^{g}/\Gamma_\tau}\) of X if and only if the Weierstrass semigroup at P is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor D0, we express the relation between the Riemann constant \({\Delta}\) and a half period in the non-symmetric case. We point out an application to an algebraic expression for the Jacobi inversion problem. We also identify the semi-canonical divisor D0 for trigonal pointed curves, namely with total ramification at P.

Light-weight implementation options for curve-based cryptography: HECC is also ready for RFID

Abstract: In this paper we describe a low footprint implementation of HyperElliptic Curve Cryptography (HECC) for RFID tags. This HECC processor supports divisor multiplication on a hyperelliptic curve defined over GF(283). We propose a Unified GF(2m) Multiplier/Inverter (UMI) which is smaller than ALUs with separated multipliers and inverters. With the UMI divisor multiplications using affine coordinates can be efficiently supported. Since affine coordinates require less registers thanprojective coordinates, the size of register file is also reduced. We choose hyperelliptic curves defined with the h(x) = x and f(x) = x5 + f 3 x3 + x2 + f 0 . The HECC processor, synthesized with 130 nm standard cell library, uses 14.5 kGates. It consumes 13.4 μW when running at 300 kHz. One divisor multiplication takes 450 ms, which makes our solution a feasible option for light-weight applications.

On the canonical decomposition of generalized modular functions

Abstract: The authors have conjectured (\cite{KoM}) that if a normalized generalized modular function (GMF) $f$, defined on a congruence subgroup $\Gamma$, has integral Fourier coefficients, then $f$ is classical in the sense that some power $f^m$ is a modular function on $\Gamma$. A strengthened form of this conjecture was proved (loc cit) in case the divisor of $f$ is \emph{empty}. In the present paper we study the canonical decomposition of a normalized parabolic GMF $f = f_1f_0$ into a product of normalized parabolic GMFs $f_1, f_0$ such that $f_1$ has \emph{unitary character} and $f_0$ has \emph{empty divisor}. We show that the strengthened form of the conjecture holds if the first "few" Fourier coefficients of $f_1$ are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either $f_0=1$ or if the divisor of $f$ is concentrated at the cusps of $\Gamma$.

Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.

Abstract: Consider the divisor sum $$\sum _{n\le N}\tau (n^2+2bn+c)$$ for integers b and c. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of $$D(-1)$$ -quadruples.