EIGENCURVE Meaning and
Definition
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The term "eigencurve" is a concept that belongs to the realm of mathematics, specifically within the field of number theory and algebraic geometry. The idea of an eigencurve was first introduced by Robert Coleman in 1996, and it has since gained significant attention among researchers working on topics related to modular forms and Galois representations.
In simple terms, an eigencurve is a mathematical construction that describes a family of modular forms associated with a fixed level and a prime number. It represents a curve in the space of modular forms, where each point on the curve corresponds to a unique eigenform.
Eigenforms are special types of modular forms that have the property of being preserved under certain transformations. The eigencurve captures the behavior of these eigenforms as the prime number varies, providing a geometric representation that allows researchers to study the distribution and properties of these forms in a unified manner.
The eigencurve is closely related to the $p$-adic Galois representations associated with the modular forms. It connects the world of modular forms to that of $p$-adic numbers and provides insights into important questions in number theory, such as the study of congruences and the behavior of modular forms modulo prime powers.
Overall, the eigencurve is a powerful mathematical tool that aids in the investigation of the intricate connections between modular forms, $p$-adic numbers, and Galois representations. Its study has contributed to important advancements in number theory and continues to be an active area of research.
Etymology of EIGENCURVE
The word "eigencurve" is derived from two separate terms: "eigen" and "curve".
1. "Eigen" comes from German and translates to "own" or "unique". It is commonly used in mathematics to denote a characteristic or distinctive property of an object. In particular, it is often used in linear algebra to refer to eigenvectors and eigenvalues, which are characteristic vectors and corresponding scalars associated with a linear transformation.
2. "Curve" refers to a continuous and smooth line or shape that typically moves through space. In mathematics, curves are often studied in the field of geometry or can represent the graph of a mathematical function.
When combined, the term "eigen" and "curve" in mathematics refers to a special kind of curve or a family of curves that possess certain characteristic properties related to the theory of modular forms and p-adic modular forms.
