This article investigates potential research pathways toward proving the Riemann Hypothesis (RH) by leveraging the mathematical structures found in arXiv:XXXXXXXXX. The paper's focus on quaternion algebras, isogeny graphs, and ideal connections offers intriguing avenues for exploring previously unconsidered relationships with the zeta function.
The paper presents equations relating the trace and norm of elements in quaternion algebras:
Trd(α) = α + ᾱ = 2x₁
Nrd(α) = αᾱ = x₁² + qx₂² + p(x₃² + qx₄²)
Potential Connection to RH: The norm form bears resemblance to certain L-functions associated with zeta function zeros. The trace relation might provide a new analytical tool for studying zero distribution.
The paper introduces connecting ideals between endomorphism rings of elliptic curves, represented as:
I = O'⋅ι'(a)
where φ_a: E' → E is an isogeny.
Potential Connection to RH: These structures could have analogies in adelic formulations of the zeta function, potentially offering insights into the distribution of primes.
The concept of O-orientations is defined as:
ι(O) = ι(K) ∩ End(E)
Potential Connection to RH: This framework could provide a new lens to study arithmetic progressions related to prime numbers and their connection to the zeta function's zeros.
This approach proposes constructing L-functions based on quaternion algebra norms:
L(s) = Σ Nrd(α)^(-s)
The zeros of these L-functions would be analyzed and linked to the Riemann zeta function through appropriate morphisms. This approach's limitations include its potential applicability to only specific L-function classes and the need for proving equivalence with classical formulations.
This approach leverages isogeny graphs, where vertices represent potential zeta zeros. The connecting ideals analyze relationships between zeros, and norm equations study spacing properties. This approach requires developing rigorous methods for mapping isogeny graph structures to the complex plane.
The research agenda includes proving foundational theorems connecting quaternion norms to zeta values and developing explicit formulas linking isogeny classes to zeta zeros. Key conjectures would need to be formulated and rigorously tested through computational experiments.
Conjecture 1: For a quaternion algebra B_p,∞, there exists a map φ: B_p,∞ → C such that zeta function zeros correspond to special values of Nrd(φ(α)).
Computational verification would involve implementing algorithms to compute quaternion norms for small cases and comparing their distribution with known zeta zeros. This research requires proving deep connections between quaternion algebras and complex analysis, bridging discrete and continuous structures, and developing new tools for L-function analysis.