Arithmetic gauge theory: A brief introduction

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

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Computational Aspects of Modular Forms and Galois Representations

10.23943/princeton/9780691142012.001.0001 ◽

2011 ◽

Author(s):

Robin de Jong ◽

Franz Merkl ◽

Johan Bosman

Keyword(s):

Polynomial Time ◽

Modular Forms ◽

Fourier Coefficients ◽

Galois Representations ◽

Arithmetic Geometry ◽

Major Step ◽

Langlands Program ◽

Special Cases ◽

Computational Aspects ◽

Fixed Power

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The book begins with a concise and concrete introduction that makes it accessible to readers without an extensive background in arithmetic geometry, and it includes a chapter that describes actual computations.

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Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

Forum of Mathematics Pi ◽

10.1017/fmp.2021.3 ◽

2021 ◽

Vol 9 ◽

Author(s):

Pierrick Bousseau ◽

Honglu Fan ◽

Shuai Guo ◽

Longting Wu

Keyword(s):

Elliptic Curve ◽

Moduli Spaces ◽

Topological String ◽

Holomorphic Anomaly ◽

Holomorphic Anomaly Equation ◽

Anomaly Equation ◽

Witten Theory ◽

Generating Series ◽

Maximal Contact ◽

Higher Genus

Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.

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Uniformizing the Moduli Stacks of Global G-Shtukas

International Mathematics Research Notices ◽

10.1093/imrn/rnz223 ◽

2019 ◽

Cited By ~ 1

Author(s):

E Arasteh Rad ◽

Urs Hartl

Keyword(s):

Finite Type ◽

Moduli Spaces ◽

Function Field ◽

Group Scheme ◽

Base Change ◽

Generic Fiber ◽

Smooth Projective Curve ◽

Langlands Program ◽

Moduli Stacks ◽

Divisible Groups

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

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On some dimension formula for automorphic forms of weight one I

Nagoya Mathematical Journal ◽

10.1017/s0027763000019711 ◽

1982 ◽

Vol 85 ◽

pp. 213-221 ◽

Cited By ~ 4

Author(s):

Toyokazu Hiramatsu

Keyword(s):

Fuchsian Group ◽

Automorphic Forms ◽

Galois Representations ◽

The Other ◽

Dimension Formula ◽

Linearly Independent ◽

Other Hand ◽

Algebraic Properties ◽

Lecture Notes ◽

Weight One

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0. But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

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Galois representations attached to automorphic forms on over fields

Compositio Mathematica ◽

10.1112/s0010437x13007665 ◽

2014 ◽

Vol 150 (4) ◽

pp. 523-567 ◽

Cited By ~ 9

Author(s):

Chung Pang Mok

Keyword(s):

Automorphic Forms ◽

Galois Representations ◽

Suitable Condition ◽

Two Dimensional ◽

Automorphic Representations ◽

Cuspidal Automorphic Representations ◽

Central Characters

AbstractIn this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

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Universal families on moduli spaces of principal bundles on curves

International Mathematics Research Notices ◽

10.1155/imrn/2006/80641 ◽

2006 ◽

Cited By ~ 4

Author(s):

V. Balaji ◽

I. Biswas ◽

D. S. Nagaraj ◽

P. E. Newstead

Keyword(s):

Moduli Spaces ◽

Principal Bundles

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A Torelli theorem for moduli spaces of principal bundles on curves defined over ℝ

International Journal of Mathematics ◽

10.1142/s0129167x17500495 ◽

2017 ◽

Vol 28 (06) ◽

pp. 1750049

Author(s):

Indranil Biswas ◽

Olivier Serman

Keyword(s):

Algebraic Group ◽

Moduli Space ◽

Moduli Spaces ◽

Isomorphism Class ◽

Projective Curve ◽

Real Numbers ◽

Principal Bundles ◽

Smooth Projective Curve ◽

Simple Factor ◽

Torelli Theorem

Let [Formula: see text] be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let [Formula: see text] be a connected reductive affine algebraic group, defined over [Formula: see text], such that [Formula: see text] is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal [Formula: see text]-bundles on [Formula: see text] determine uniquely the isomorphism class of [Formula: see text].

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Oddness of residually reducible Galois representations

International Journal of Number Theory ◽

10.1142/s1793042118500835 ◽

2018 ◽

Vol 14 (05) ◽

pp. 1329-1345 ◽

Cited By ~ 1

Author(s):

Tobias Berger

Keyword(s):

Automorphic Forms ◽

Galois Representations ◽

Selmer Group ◽

Parity Condition

We show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the ±-Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this, we relate the oddness of the associated polarized Galois representations (in the sense of the Bellaïche-Chenevier sign being [Formula: see text]) to the parity condition for criticality. Under an assumption similar to Vandiver’s conjecture this also provides evidence for the Fontaine–Mazur conjecture for residually reducible polarized Galois representations.

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