On invariants and the theory of numbers 论不变量与数论

Of enormous historical importance, this classic offered the first public formulation of Dickson's theory of invariants for modular forms and linear transformations. In many sections of the five lectures included here, Dickson aimed not at complete generality, but at an illumination of typical and suggestive topics. The introductory lecture is followed by sections on seminvariants of algebraic and modular binary forms; invariants of a modular group and formal invariants and covariants of modular forms; modular geometry and covariantive theory of a quadratic form in m variables, modulo 2; and a theory of plane cubic curves with a real inflexion point valid in ordinary and in modular geometry. 1914 ed.

INTRODUCTION
LECTURE I
A THEORY OF INVARNTTS APPLICABLE TO ALGEBRAIC AND MODULAR FORMS
1-3. Introduction to the algebraic side of the theory by means of the example of an algebraic quadratic form in m variables
4-7. Introduction to the number theory side of the theory of invariants by means of the example of a modular quadratic form
8-9. Modular invariants are rational and integral
10. Characteristic modular invariants
11. Number of linearly independent modular invariants
12. Fundamental system of modular invariants
13. Minor ro1e of modular covariants
14. References to further developments
LECTURE II
SEMINVARIANTS OF ALGEBRAIC AND MODULAR BINARY FORMS
1-6. Introductory example of the binary quartic form
7-10. Fundamental system of modular seminvariants of a binary n-ic derived by induction from n - 1 to n
11. Explicit fundamental system when the modulus exceeds n
12. Another method for the ease p > n
13. Number of linearly independent seminvariants
14-15. Derivation of modular invariants from semin-variants
LECTURE III INVARIANTS OF A MODULAR GROUP. FORMAL INVARIANTS AND COVARIANTS OF MODULAR FORMS. APPLICATIONS
1-4. Invariants of certain modular groups; problem of Hurwitz
5-11. Formal invariants and scminvariants of binary modular forms
12. Theorem of Miss Sanderson
13. Fundamental systems of modular covariants
14. Form problem for the total binary modular group
15. Invariantive classification of forms
LECTURE IV MODULAR GEOMETRY AND COYARIANTIVE THEORY OF A QUAD RATIC FORM IN m VARIABLES MODULO 2
1-2. Introduction. The polar locus
3. Odd number of variables; apex; linear tangential equation
4. Covariant line of a conic
5. Even number of variables
6. Covariant plane of a degenerate quadric surface
7. A configuration defined by the quinary surface
8. Certain formal and modular covariants of a conic
9-32. Fundamental system of covariants of a conic
33. References on modular geometry
LECTURE V A THEORY OF PLANE CUBIC CURVES WITH A REAL INFLEXION POIN VALID IN ORDINARY AND IN MODULARGEOMETRY