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

2016 ◽  
Vol 13 (01) ◽  
pp. 39-63 ◽  
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].

scholarly journals The arithmetic of vector-valued modular forms on Γ0(2)

2019 ◽  
Vol 16 (02) ◽  
pp. 241-289
Author(s):  
Richard Gottesman
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Fourier Coefficients ◽  
Weight Vector ◽  
Free Module ◽  
Two Dimensional ◽  
Dimensional Representation ◽  
Minimal Weight ◽  
Gaussian Hypergeometric Series ◽  
Vector Valued

Let [Formula: see text] denote an irreducible two-dimensional representation of [Formula: see text] The collection of vector-valued modular forms for [Formula: see text], which we denote by [Formula: see text], form a graded and free module of rank two over the ring of modular forms on [Formula: see text], which we denote by [Formula: see text] For a certain class of [Formula: see text], we prove that if [Formula: see text] is any vector-valued modular form for [Formula: see text] whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of [Formula: see text] is unbounded. Our methods involve computing an explicit basis for [Formula: see text] as a [Formula: see text]-module. We give formulas for the component functions of a minimal weight vector-valued form for [Formula: see text] in terms of the Gaussian hypergeometric series [Formula: see text], a Hauptmodul of [Formula: see text], and the Dedekind [Formula: see text]-function.

scholarly journals Atiyah’s work on holomorphic vector bundles and gauge theories

2021 ◽  
Vol 58 (4) ◽  
pp. 567-610
Author(s):  
Simon Donaldson
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Vector Bundles ◽  
Gauge Theories ◽  
Holomorphic Vector Bundles ◽  
And Topology ◽  
The 1950S

The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

scholarly journals Mazur’s rational torsion result for pointless genus one curves: examples

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Arjan Dwarshuis ◽  
Majken Roelfszema ◽  
Jaap Top
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Rational Point ◽  
Torsion Points ◽  
Arbitrary Genus ◽  
Mazur’S Theorem ◽  
Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

scholarly journals Coherent state transforms and vector bundles on elliptic curves

2003 ◽  
Vol 204 (2) ◽  
pp. 355-398 ◽  
Author(s):  
Carlos A. Florentino ◽  
José M. Mourão ◽  
João P. Nunes
Keyword(s):  
Coherent State ◽  
Elliptic Curves ◽  
Vector Bundles

scholarly journals A trace formula for vector-valued modular forms

2015 ◽  
Vol 17 (06) ◽  
pp. 1550069
Author(s):  
P. Bantay
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Weight Distribution ◽  
Hilbert Polynomial ◽  
Free Generators ◽  
Vector Valued

We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

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