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Introduction to Algebraic and Abelian Functions is a self-contained presentation of a fundamental subject in algebraic geometry and number theory. For this revised edition, the material on theta functions has been expanded, and the example of the Fermat curves is carried throughout the text. This volume is geared toward a second-year graduate course, but it leads naturally to the study of more advanced books listed in the bibliography.

Inhaltsverzeichnis

Frontmatter

Chapter I. The Riemann-Roch Theorem

Abstract
We recall that a discrete valuation ring o is a principal ideal ring (and therefore a unique factorization ring) having only one prime. If t is a generator of this prime, we call t a local parameter.
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Chapter II. The Fermat Curve

Abstract
The purpose of this chapter is to give a significant example for the notions and theorems proved in the first chapter.
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Chapter III. The Riemann Surface

Abstract
The purpose of this chapter is to show how to give a structure of analytic manifold to the set of points on a curve in the complex numbers, but our treatment also applies to more general fields like p-adic numbers. We are principally interested in the complex case, in order to derive the Abel-Jacobi theorem in the next chapter.
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Chapter IV. The Theorem of Abel-Jacobi

Abstract
A differential will be said to be of the first kind if it is holomorphic everywhere on the Riemann surface. Such differentials form a vector space over the complex, and by the Riemann-Roch theorem, one sees that the dimension of this space is equal to the genus g of R.
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Chapter V. Periods on the Fermat Curve

Abstract
We return to the special case of the Fermat curve, and compute the period lattice explicitly in terms of the basis for the differentials of first kind given in Chapter II.
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Chapter VI. Linear Theory of Theta Functions

Abstract
Let V be a complex vector space of dimension n, real dimension 2n. Let D be a lattice in V, that is, a discrete subgroup of real dimension 2n, so that the factor group V/D is a complex torus. We define a theta function on V, with respect to D (or on V/D), to be a quotient of entire functions (called a meromorphic function for this chapter), not identically zero, and satisfying the relation
$$F(x + u) = F(x){e^{2\pi i[L(x,u) + J(u)]}},{\text{ }}all{\text{ }}x{\text{ }} \in {\text{V,u}} \in {\text{D}}$$
(1)
where L is C-linear in x, and no specifications are made on its dependence on u, or on the dependence of the function J on u. However, we note that we can change J by a Z-valued function on D without changing the above equation. Also, we shall see below that any such L and J must satisfy additional conditions which can be deduced from this equation. We note that the theta functions form a multiplicative group.
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Chapter VII. Homomorphisms and Duality

Abstract
This chapter describes the elementary theory of homomorphisms and endomorphisms of an abelian manifold. First we relate the rational and complex representations to a purely algebraic representation on the points of finite order. Then we prove the complete reducibility theorem of Poincaré, showing that an abelian manifold admits a product decomposition into simple ones, up to isogeny. Finally, we deal with the duality which arises from the nondegenerate hermitian form, and show how the dual manifold corresponds to divisor classes of divisors algebraically equivalent to 0. The duality includes an essentially algebraic pairing between points and such divisors, and a formula in the last section relates this algebraic pairing with the analytic data and the Riemann form.
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Chapter VIII. Riemann Matrices and Classical Theta Functions

Abstract
Let H be a positive definite hermitian form on C n . We may write H in terms of its real and imaginary parts asH(u, v) = E(iu, v) + iE(u, v) where E is alternating and real valued.
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Chapter IX. Involutions and Abelian Manifolds of Quaternion Type

Abstract
Certain abelian manifolds have large algebras of endomorphisms. The most common case is that of Complex Multiplication, which is treated extensively in the literature. Almost as important is the case when this algebra contains a quaternion algebra. I have therefore included this section as an example of such manifolds, which will provide easier access to their more advanced theory.
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Chapter X. Theta Functions and Divisors

Abstract
Let M be a complex manifold. In the sequel, M will either be Cn or Cn/D where D is a lattice (discrete subgroup of real dimension 2n). Let U i be an open covering of M, and let ϕi be a meromorphic function on Ui. If for each pair of indices (i, j) the function ϕi /ϕj is holomorphic and invertible on UiUj, then we shall say that the family (Ui, ϕi) represents a divisor on M. If this is the case, and (U, ϕ) is a pair consisting of an open set U and a meromorphic function ϕ on U, then we say that (U, ϕ) is compatible with the family (Ui, ϕi) if ϕi/ϕ is holomorphic invertible on UUi. If this is the case, then the pair (U, ϕ) can be adjoined to our family, and again represents a divisor. Two families (Ui, ϕi) and (Vk, ψk) are said to be equivalent if each pair (Vk, ψk) is compatible with the first family. An equivalence class of families as above is called a divisor on M. Each pair (U, ϕ) compatible with the families representing the divisor is also said to represent the divisor on the open set U.
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Backmatter

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