Heegner number

Heegner Number

A Heegner number is a specific type of integer that plays a significant role in number theory, particularly in the context of complex multiplication and the theory of elliptic curves. These numbers are named after the German mathematician Kurt Heegner, who first identified their significance in the mid-20th century.

Definition and Properties

A Heegner number is a positive integer \( d \) such that the imaginary quadratic field \( \mathbb{Q}(\sqrt{-d}) \) has class number one. In simpler terms, these are the values of \( d \) for which the ring of integers in the corresponding imaginary quadratic field has a unique factorization property, meaning every ideal in this ring is principal.

The known Heegner numbers are: \[ 1, 2, 3, 7, 11, 19, 43, 67, 163 \]

These numbers are exceptional because they are the only values of \( d \) for which the class number of \( \mathbb{Q}(\sqrt{-d}) \) is one. This property was proven rigorously by Alan Baker and Harold Stark in the 1960s, confirming Heegner's earlier work.

Historical Context

Kurt Heegner published his findings in 1952, but his work was initially met with skepticism due to some gaps in the proof. It wasn't until Baker and Stark independently verified and completed Heegner's proof in the 1960s that his contributions were fully recognized. This breakthrough was significant in the field of algebraic number theory.

Applications in Number Theory

Heegner numbers have profound implications in various areas of mathematics:

Elliptic Curves

In the theory of elliptic curves, Heegner numbers are related to the complex multiplication of elliptic curves. The j-invariant of an elliptic curve with complex multiplication by the ring of integers of \( \mathbb{Q}(\sqrt{-d}) \) is an algebraic integer. These j-invariants are often used to construct class fields over imaginary quadratic fields.

Class Field Theory

Heegner numbers are also crucial in class field theory. The fields \( \mathbb{Q}(\sqrt{-d}) \) for these values of \( d \) are the simplest examples of fields with class number one. This property simplifies the study of their arithmetic and the construction of their maximal abelian extensions.

Modular Functions

The values of modular functions at Heegner points (points on the upper half-plane that are quadratic irrationalities) are algebraic numbers. These values are used in the explicit class field theory of imaginary quadratic fields.

Proof and Verification

The proof of the finiteness of Heegner numbers involves several advanced concepts in number theory:

Baker's Method

Alan Baker used techniques from transcendental number theory to provide bounds on linear forms in logarithms of algebraic numbers. This method was instrumental in proving the finiteness of Heegner numbers.

Stark's Contribution

Harold Stark provided a different approach using analytic methods, particularly involving the properties of L-functions associated with imaginary quadratic fields. His work complemented Baker's results and helped establish the completeness of Heegner's original proof.

Further Implications

The uniqueness of Heegner numbers has led to further research in related areas:

Generalizations

Mathematicians have explored generalizations of Heegner numbers to other quadratic fields and higher-dimensional analogs. These investigations often involve deep results from arithmetic geometry and the theory of automorphic forms.

Computational Aspects

The identification and verification of Heegner numbers have also spurred developments in computational number theory. Algorithms for computing class numbers and verifying the properties of quadratic fields have been refined and optimized.

References