Galois representations

Definition

Galois representations are mathematical objects that encode the action of a Galois group on a vector space, typically associated with algebraic objects like number fields or algebraic varieties. These representations allow for the study of symmetries in arithmetic, relating number theory and geometry through various structures such as modular forms and L-functions.

5 Must Know Facts For Your Next Test

1. Galois representations can be viewed as homomorphisms from the Galois group of a field extension to a linear algebraic group, providing deep insights into the structure of the field.
2. They often arise in the study of elliptic curves, where the action of the Galois group can reveal properties of the curve's points over various fields.
3. In the context of modular forms, Galois representations help in understanding how these forms transform under the action of Galois groups, linking them to number-theoretic properties.
4. The Langlands program proposes deep connections between Galois representations and automorphic forms, suggesting a framework for understanding their interactions.
5. Galois representations are crucial in proving results like Serre's conjecture, which posits that every odd, irreducible two-dimensional representation over a finite field arises from a modular form.

Review Questions

• How do Galois representations connect to modular forms and what implications does this have for number theory?
  • Galois representations provide a way to understand how modular forms transform under the action of Galois groups. This connection is significant because it allows mathematicians to use properties of modular forms to draw conclusions about the corresponding Galois representations. For example, results like the Modularity Theorem link these representations to elliptic curves, showing that many elliptic curves can be associated with specific modular forms.
• Discuss the role of Galois representations in the context of Serre's modularity conjecture and its implications for number theory.
  • Serre's modularity conjecture states that every odd, irreducible two-dimensional representation over a finite field should arise from a modular form. This conjecture highlights the importance of Galois representations in connecting different areas of mathematics, as proving this conjecture would establish deep links between number theory and geometry. The implication is that understanding these representations can lead to insights into the distribution of prime numbers and properties of arithmetic objects.
• Evaluate how the Langlands program uses Galois representations to propose connections between different areas of mathematics, specifically in terms of symmetry and duality.
  • The Langlands program suggests profound connections between Galois representations and automorphic forms, indicating that there is a duality between them. This duality reflects symmetries in mathematical structures, allowing insights from one area to inform another. For instance, understanding how Galois groups act on various spaces via these representations helps link number theory with harmonic analysis, ultimately providing a unifying framework for studying mathematical phenomena across different disciplines.

Related terms

L-functions: These are complex functions associated with number-theoretic objects that encode important information about their properties, often connected to Galois representations through Langlands program.

Modular forms: These are complex functions that are invariant under the action of a certain group and play a crucial role in connecting Galois representations to number theory.

Galois groups: These groups capture the symmetries of field extensions and play a central role in defining Galois representations, linking them to polynomial roots and algebraic structures.