Vector Bundles on Elliptic Curves

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite fieldFqand any integern≥1bywhere the sum is over isomorphism classes ofFq-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function ofX/Fqifn=1, is a rational function ofq−swith denominator(1−q−ns)(1−qn−ns)and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet serieswhere the sum is now over isomorphism classes ofFq-rational semistable vector bundles V of degree 0 on X, is equal to∏k=1∞ζX/Fq(s+k),and use this fact to prove the Riemann hypothesis forζX,n(s)for all n.

Download Full-text

Vector bundles on the moduli stack of elliptic curves

Journal of Algebra ◽

10.1016/j.jalgebra.2015.01.007 ◽

2015 ◽

Vol 428 ◽

pp. 425-456 ◽

Cited By ~ 4

Author(s):

Lennart Meier

Keyword(s):

Elliptic Curves ◽

Vector Bundles ◽

Moduli Stack

Download Full-text

Linear Codes Obtained from Projective and Grassmann Bundles on Curves

Contemporary Mathematics ◽

10.37256/cm.142020449 ◽

2020 ◽

Author(s):

Edoardo Ballico

Keyword(s):

Vector Bundle ◽

Elliptic Curves ◽

Vector Bundles ◽

Linear Codes ◽

Smooth Curve ◽

Set Up ◽

Semistable Vector Bundle ◽

Double Coverings

We use split vector bundles on an arbitrary smooth curve defined over Fq to get linear codes (following the general set-up considered by S. H. Hansen and T. Nakashima), generalizing two quoted results by T. Nakashima. If p ≠ 2 for all integers d, g ≥ 2, r > 0 such that either r is odd or d is even we prove the existence of a smooth curve C of genus g defined over Fq and a p-semistable vector bundle E on C such that rank(E) = r, deg(E) = d and E is defined over Fq. Most results for particular curves are obtained taking double coverings or triple coverings of elliptic curves.

Download Full-text

Vector-valued modular forms and the modular orbifold of elliptic curves

International Journal of Number Theory ◽

10.1142/s179304211750004x ◽

2016 ◽

Vol 13 (01) ◽

pp. 39-63 ◽

Cited By ~ 7

Author(s):

Luca Candelori ◽

Cameron Franc

Keyword(s):

Algebraic Geometry ◽

Elliptic Curves ◽

Modular Forms ◽

Vector Bundles ◽

Projective Line ◽

Free Module ◽

Weighted Projective Line ◽

Holomorphic Vector Bundles ◽

Vector Valued ◽

Standard Techniques

This paper presents the theory of holomorphic vector-valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector-valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of [Formula: see text] play in the holomorphic theory of vector-valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line [Formula: see text].

Download Full-text