Admissible representations

Definition

Admissible representations are a specific type of representation in the context of p-adic representations that satisfy certain finiteness conditions. They are crucial in understanding the structure of Galois representations and play a significant role in bridging different areas of mathematics, particularly when connecting p-adic Hodge theory and local class field theory. These representations help in analyzing how various mathematical objects behave under the action of Galois groups.

5 Must Know Facts For Your Next Test

1. Admissible representations are often constructed from continuous linear representations of Galois groups over p-adic fields.
2. They have an important role in the study of modular forms, where their properties can help understand the associated L-functions.
3. In p-adic Hodge theory, admissible representations allow for a detailed comparison between the topology of p-adic spaces and algebraic structures.
4. Admissible representations can be characterized by their behavior under various types of completions and their compatibility with local fields.
5. These representations frequently come into play when studying the local Langlands correspondence, which connects Galois representations to automorphic forms.

Review Questions

• How do admissible representations connect p-adic Hodge theory with Galois representations?
  • Admissible representations are integral to p-adic Hodge theory as they allow mathematicians to analyze the relationships between Galois groups and various algebraic structures. By satisfying certain finiteness conditions, these representations help in translating problems in algebraic geometry into problems concerning p-adic analysis. This connection provides insights into both areas and facilitates a deeper understanding of their interplay.
• Discuss how admissible representations relate to the local Langlands correspondence and its implications.
  • Admissible representations play a significant role in the local Langlands correspondence, which seeks to establish connections between Galois representations and automorphic forms. In this correspondence, admissible representations help classify and understand how various representations behave under Galois actions. This relationship is key to developing tools for number theory and representation theory, allowing insights into broader mathematical phenomena.
• Evaluate the importance of admissible representations in the context of local class field theory, considering their applications.
  • Admissible representations are vital in local class field theory as they facilitate the study of abelian extensions of local fields through Galois groups. Their properties ensure that various algebraic structures maintain compatibility with field extensions, leading to better insights into class numbers and local reciprocity laws. Understanding these connections allows mathematicians to derive significant results regarding the behavior of units and ideals within local fields, making them indispensable in this area of research.

Related terms

p-adic numbers: Numbers that extend the ordinary arithmetic of the integers and rational numbers with respect to a prime number p, leading to a new system of analysis and number theory.

Galois group: A group associated with a field extension that describes the symmetries and automorphisms of the extension, important in understanding the roots of polynomials.

Finiteness condition: A property that restricts the size or dimension of an object, often used to ensure certain desirable features in mathematical structures.