Galois Theory and the Riemann Hypothesis: New Research Pathways

Introduction

This article outlines potential research pathways for approaching the Riemann Hypothesis. It uses mathematical frameworks, techniques, and perspectives from the paper arXiv:hal-03739451v3 to tackle this long-standing problem.

Mathematical Frameworks

Galois Cohomology and Class Field Theory

The paper uses sequences related to Galois cohomology and class field theory, offering insights into the relationships between Galois groups, ideal class groups, unit groups, and norm maps in field extensions. For example:

1 → Ker(J_L/K) → H¹(G, E_L) → Coker(j_L/K) → Coker(J_L/K) → H²(G, E_L) → H²(G, L^×)

These structures can be used to study the connection between cohomology groups and L-functions and relate the paper's group cohomology sequence to functional equations of L-functions.

Norm Map Framework

The paper uses exact sequences involving norm maps, such as:

1 → N H → H → N_L/K(H) → 1

This framework can be used to investigate norm relationships in field extensions and potentially study zeros of zeta functions through norm maps.

Novel Approaches

Cohomological Height Approach

Using structures from the paper, such as:

#(H_Lⁱ⁺¹ / H_Lⁱ) = (# H_K)/(# N_L/K(H_Lⁱ)) × p^n(r-1)/(Λ_Kⁱ : Λ_Kⁱ ∩ N_L/K(L^×))

Research could focus on studying height relationships in field extensions and investigating how these relate to zeta function properties, developing analogous formulations for the Riemann zeta function.

Analyzing Algebraic Structures and Zeta Zero Distribution

Explore how modifications in algebraic structures influence the hypothetical symmetry in the distribution of zeros, using sequences such as (α_M^s/(1-ζ_ℓ)^r_s)^σ. Define algebraic modifications in terms of deformation of rings and fields, and analyze how these deformations affect the properties of the zeta function, particularly around critical lines.

Tangential Connections

Elliptic Curves and Modular Forms

Use the modularity theorem to relate elliptic curves to modular forms, and connect modular forms to the Riemann zeta function. Explore potential connections between the Galois cohomology of L/K and the elliptic curves over K.

Ergodic Theory and the Distribution of Zeros

The distribution of zeros of the Riemann zeta function can be studied using techniques from ergodic theory. The statistical properties of the kernels and cokernels in the Galois cohomology sequences are related to the statistical properties of the spacing between zeros of the Riemann zeta function.

Research Agenda

Intermediate Results to Establish

Develop cohomological criteria for zero distribution.
Establish norm map relationships to critical line properties.
Prove finite-degree cases first.

Required Tools

Group cohomology techniques from the paper.
Class field theory methods.
L-function analysis tools.

Computational Verification

Study simplified cases using (α_M^s/(1-ζ_ℓ)^r_s)^σ.

Conclusion

This structured approach integrates intricate algebraic and topological concepts from arXiv:hal-03739451v3, aiming to create a pathway toward proving the Riemann Hypothesis by connecting these areas with the fundamental properties of zeta functions.