**Colloquium Publications**

Volume: 4;
1914;
238 pp;
Softcover

MSC: Primary 11; 32;

Print ISBN: 978-0-8218-4598-1

Product Code: COLL/4

List Price: $58.00

AMS Member Price: $46.40

MAA Member Price: $52.20

**Electronic ISBN: 978-1-4704-3154-9
Product Code: COLL/4.E**

List Price: $58.00

AMS Member Price: $46.40

MAA Member Price: $52.20

# The Madison Colloquium

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*Leonard Eugene Dickson; William Fogg Osgood*

Following the tradition of the American Mathematical Society, the seventh
colloquium of the Society was held as part of the summer meeting that took
place at the University of Wisconsin, in Madison. Two sets of lectures
were presented: On Invariants and the Theory of Numbers, by L. E.
Dickson, and Functions of Several Complex Variables, by W. F.
Osgood.

Dickson considers invariants of quadratic forms, with a special
emphasis on invariants of forms defined in characteristic
\(p\), also called modular invariants, which have
number-theoretic consequences. He is able to find a fundamental set of
invariants for both settings. For binary forms, Dickson introduces
semi-invariants in the modular case, and again finds a fundamental
set. These studies naturally lead to the important study of invariants
of the standard action of the modular group. The lectures conclude
with a study of “modular geometry”, which is now known as
geometry over \(\mathbf{F}_p\).

The lectures by Osgood review the state of the art of several complex
variables. At this time, the theory was entirely function-theoretic.
Already, though, Osgood can introduce the ideas and theorems that will be
fundamental to the subject for the rest of the century: Weierstrass
preparation, periodic functions and theta functions,
singularities—including Hartogs' phenomenon, the boundary of a domain of
holomorphy, and so on.

#### Readership

Graduate students and research mathematicians interested in number theory and analysis.

#### Table of Contents

# Table of Contents

## The Madison Colloquium

- Cover Cover11
- Title page i2
- Preface iii4
- L. E. Dickson. On Invariants and the Theory of Numbers. vii8
- Contents ix10
- Introduction 114
- Lecture I: A theory of invariants applicable to algebraic and modular forms 417
- Lecture II: Seminvariants of algebraic and modular binary forms 1629
- Lecture III: Invariants of a modular group. Formal Invariants and covariants of modular forms. Applications 3346
- Lecture IV: Modular geometry and covariantive theory of a quadratic form in 𝑚 variables modulo 2 6578
- Lecture V: A theory of plane cubic curves with a real inflexion point valid in ordinary and in modular geometry 99112
- W. F. Osgood. Topics in the Theory of Functions of Several Complex Variables. A-i124
- Contents B-i126
- Lecture I: A General Survey of the Field 111130
- Lecture II: Some General Theorems 133152
- Lecture III: Singular Points and Analytic Continuation 160179
- Lecture IV: Implicit Functions 181200
- Lecture V: The Prime Function on an Algebraic Configuration 199220
- Back Cover Back Cover1252