Unlocking Riemann: Elliptic Curves and Prime Distribution

Detailed Research Pathways for Approaching the Riemann Hypothesis

Framework Identification and Application

1. Elliptic Curve Interaction with Zeta Function:

   • Mathematical Framework: Utilize the structures from the elliptic curves (E_{1}^{\prime} \times E_{2}^{\prime} and similar formulations) and their interactions as defined by the pairing functions in the paper to model properties of the zeta function.
   • Formulation: Consider the pairing e_{E_{1} \times E_{2}, n}(P, Q) and explore its implications on the analytic properties of the zeta function, especially focusing on the zeros in the critical strip.
   • Theorem Construction: Prove that the behavior of these pairings under certain transformations aligns with the non-trivial zeros of the zeta function. Establish a formal link via L-functions associated with these curves.

2. Degeneration Methods in Endomorphism Rings:

   • Mathematical Framework: Use the concept of degeneration in endomorphism rings to study the shifts in zeros of modular forms, which are deeply linked to the zeta function through their L-series.
   • Formulation: Define \operatorname{deg}(\alpha_{0}) = z_{0}^{2} f where \alpha_{0} is an endomorphism, and examine its impact on the spectral decomposition of the Laplace operator on modular curves.
   • Theorem Construction: Develop lemmas that connect the degeneration of endomorphism rings to the distribution of zeros of the corresponding modular forms' L-series, and by extension, to the zeta function.

3. Computational Algebraic Geometry:

   • Mathematical Framework: Apply computational methods in algebraic geometry to the tabulated complex structures related to the zeta function.
   • Formulation: Employ algorithms for computing cohomologies of the varieties defined by the relations in the tabular data, analyzing their impact on the critical line conjecture.
   • Theorem Construction: Show that the computational results align with the Riemann hypothesis predictions for specific families of algebraic varieties.

Novel Combined Approaches

1. Combining Elliptic Curves and Explicit Formulae Approach:

   • Mathematical Foundation: Integrate explicit formulae relating the zeros of the zeta function to primes (via logarithmic derivatives) with the behavior of elliptic curves over finite fields as described in the paper.
   • Methodology: Develop a comparative analysis between the growth rates of coefficients in the Fourier expansion of modular forms derived from elliptic curves and the oscillatory behavior of the zeta function on the critical line.
   • Predictions and Limitations: Predict that the alignment in growth rates and oscillations provides a new pathway to verify RH. Address potential limitations in non-generic cases or exceptional isogenies by extending the class of elliptic curves considered.

2. Endomorphism Degeneration and Zero-Free Regions:

   • Mathematical Foundation: Use the concept of endomorphism degeneration to predict zero-free regions near the critical line of the zeta function.
   • Methodology: Establish a metric on the space of endomorphisms that correlates with distances from the critical line, exploring how degeneration affects this metric.
   • Predictions and Limitations: Formulate conjectures on zero distribution based on degeneration patterns. Tackle limitations by refining the degeneration metrics and incorporating error terms from the explicit formulae.

Tangential Connections and Conjectures

1. Isogeny Action on Zeta Zeros:

   • Connection: Explore how the action of isogenies in the elliptic curve frameworks affects the conjectured symmetries in the zeros of the zeta function.
   • Conjecture: There exists an isogeny action that preserves the critical strip structure and provides insights into the symmetry about the critical line.
   • Computational Experiment: Simulate the action of various isogenies on test cases where zeros are known to high precision, and analyze the symmetry preservation.

Research Agenda

• Conjectures to Prove:
  1. Pairing behaviors in elliptic curve products directly correspond to movements of zeros of the zeta function.
  2. Degeneration in endomorphism rings predicts zero-free regions of the zeta function.
• Mathematical Tools Required: Advanced algebraic geometry tools, computational algebra systems, and analytic number theory techniques.
• Intermediate Results:
  1. Verification of pairing behavior under specific elliptic curve transformations.
  2. Statistical analysis of zero-free regions predicted by endomorphism degeneration.
• Theorem Sequence:
  1. Establish basic properties of the elliptic curve frameworks.
  2. Link these properties to modular forms and their L-series.
  3. Translate these findings to implications for the zeta function.
• Example: Demonstrate using a specific elliptic curve that its endomorphism ring's behavior under degeneration aligns with known data about zeta function zeros.

This structured approach provides a clear pathway from deep theoretical foundations to potential proof strategies for the Riemann Hypothesis, drawing inspiration from the mathematical structures detailed in arXiv:hal-04023441v1.