Modularity of PGL₂(��p)-representations over totally real fields

Area of Science:

• Number Theory
• Algebraic Geometry
• Representation Theory

Background:

• Serre's modularity conjecture relates Galois representations to modular forms.
• Projective representations generalize standard representations in abstract algebra.
• Totally real number fields are crucial in algebraic number theory.

Purpose of the Study:

• To investigate an analog of Serre's modularity conjecture for projective representations.
• To extend the understanding of the relationship between number theory and modular forms.
• To explore these concepts over totally real number fields.

Main Methods:

• Utilizing techniques from algebraic number theory.
• Applying methods related to the theory of modular forms.
• Focusing on the properties of projective representations.

Main Results:

• Proved specific cases of the analog of Serre's modularity conjecture.
• Demonstrated the conjecture's validity under certain conditions for projective representations.
• Provided new insights into the behavior of these representations over number fields.

Conclusions:

• The study successfully proves cases of the analog of Serre's modularity conjecture.
• Results contribute to the broader understanding of modularity in number theory.
• Further research can explore additional cases and related conjectures.