This analysis explores potential research pathways for attacking the Riemann Hypothesis based on mathematical frameworks presented in the paper arXiv:4557.7845. The paper focuses on isogenies of elliptic curves and related algebraic structures. While seemingly distant from the Riemann zeta function, these concepts can be connected through various number-theoretic bridges.
Mathematical Frameworks and Connections
Weil Pairing and Isogenies
The paper extensively uses the Weil pairing and isogenies between elliptic curves. A fundamental identity discussed involves a function h and its 'adjoint' or 'dual' function, often denoted as tilde-h, related through the Weil pairing. Another key relationship connects tilde-h to the dual of h (h-vee) via polarization maps (lambda functions).
- Core Idea: The duality relationship in the Weil pairing structure has structural similarity to functional equations found in number theory, including the one for the Riemann zeta function.
- Potential Theorem: Develop a bilinear form analogous to the Weil pairing for arguments related to the zeta function. Prove that this form exhibits symmetries analogous to those of the Weil pairing, potentially highlighting the critical line's role.
- Connection: The symmetrical properties observed in elliptic curve pairings and dual isogenies might provide structural insights into the functional equation and critical line symmetry of the Riemann zeta function.
Ideal Class Group and Isogenies
The paper mentions finding ideals in the endomorphism ring of an elliptic curve with a specific norm to construct isogenies. This ties into the structure of the ideal class group of the endomorphism ring.
- Core Idea: When the endomorphism ring is an order in an imaginary quadratic field, its Dedekind zeta function is related to the distribution of ideal norms. Dedekind zeta functions share properties with the Riemann zeta function.
- Connection: Studying the distribution of ideal classes and norms via isogeny construction could provide analogous insights into the zero distribution of related zeta functions, potentially transferable to the Riemann case.
Composition of Isogenies
The paper describes constructing new maps by composing isogenies, such as a map 'f' from a product of curves C and E to E0 times X, defined using combinations of points under related isogenies.
- Core Idea: Composing isogenies often corresponds to multiplying associated L-functions.
- Connection: By carefully selecting and composing isogenies, one might construct L-functions with specific zero distributions, potentially modeling or revealing properties of the Riemann zeta function's zeros.
Novel Approaches
Elliptic Curve L-functions and Isogeny Twists
Combine the study of Hasse-Weil L-functions of elliptic curves with isogeny structures.
- Mathematical Foundation: Define an 'isogeny twist' of an elliptic curve L-function as a sum of L-functions of curves related by isogenies, with carefully chosen coefficients.
- Proposed Theorem: Conjecture that there exists a specific family of isogenies and coefficients such that the zeros of the resulting isogeny-twisted L-function lie on the critical line if and only if the Riemann Hypothesis is true.
- Methodology: Start with a well-understood elliptic curve (e.g., with complex multiplication). Generate families of isogenies. Compute the L-functions of the resulting curves. Develop methods (possibly computational or using random matrix theory insights) to select coefficients. Prove analytic properties of the twisted L-function, focusing on zero distribution.
- Predictions: This approach could highlight a deep link between elliptic curve arithmetic and prime distribution, potentially showing RH is equivalent to an equidistribution property of zeros under isogeny twists.
- Limitations: Constructing and analyzing these twisted L-functions is complex. Finding the correct coefficients may require advanced techniques.
Tangential Connections
Quantum Chaos and Isogeny Graphs
Isogeny graphs of elliptic curves share statistical properties with quantum chaotic systems.
- Connection: The distribution of eigenvalues of adjacency matrices for isogeny graphs can resemble the distribution of zeros of the Riemann zeta function.
- Conjecture: A precise mathematical link exists between the spectral statistics of isogeny graphs and the distribution of Riemann zeta zeros.
- Computational Experiment: Generate large isogeny graphs, compute their spectral data, and compare statistically to known data on Riemann zeta zeros.
Modular Forms and Elliptic Curves
The Modularity Theorem connects elliptic curves to modular forms, whose L-functions have rich structures.
- Connection: The L-function of an elliptic curve is the L-function of a modular form. Properties of modular forms are deeply linked to number theory.
- Conjecture: The Riemann Hypothesis is equivalent to a specific statement about the analytic properties of L-functions of certain families of modular forms associated with elliptic curves.
- Computational Experiment: Study the zero distribution of L-functions for families of modular forms arising from elliptic curves and compare to the Riemann zeta function's zeros.
Detailed Research Agenda
Focusing on the 'Elliptic Curve L-functions and Isogeny Twists' approach:
Step 1: Formalize Isogeny Twists
- Conjecture: An elliptic curve E and a family of isogenies {phi_i: E -> E_i} exist such that the zeros of the isogeny twist L_twist(E, s) (defined as a sum of L(E_i, s) with coefficients) approach the critical line as the family grows.
- Tools: Isogeny computation algorithms, elliptic curve L-function computation, random matrix theory insights, analytic techniques.
- Intermediate Result: Prove that the zeros of L_twist(E, s) lie within a vertical strip containing the critical line.
Step 2: Link Twist Zeros to RH
- Conjecture: The Riemann Hypothesis holds if and only if there exist specific coefficients {a_i} such that all zeros of L_twist(E, s) lie precisely on the critical line.
- Tools: Explicit formulas for L-functions, harmonic analysis, analytic number theory techniques.
- Intermediate Result: Prove that the number of zeros of L_twist(E, s) on the critical line is unbounded as the family size increases.
Step 3: Prove Equivalence
- Theorem: (Assuming Conjectures above) The Riemann Hypothesis is true.
- Proof Outline: Demonstrate that the condition on the zeros of L_twist(E, s) directly implies the distribution of prime numbers required for the Riemann Hypothesis.
Example: Start with an elliptic curve with complex multiplication. Generate isogenies of prime degree. Compute L-functions. Experimentally search for coefficients that concentrate zeros on the critical line. Analytically investigate if this concentration implies RH.