The Birch–Swinnerton–Dyer Rank Problem: Reduction of the Analytic Rank Two Case

Abstract

This paper studies the Birch-Swinnerton-Dyer (BSD) conjecture for elliptic curves E defined over the rational number field Q. The central focus is the construction of a general structure arising from elliptic curves, followed by transformations that lead to a symmetry comparison between the analytic and algebraic sides. The results are established within the scope of the constructed structure and the specified symmetry-compatibility condition.

Define the analytic rank and algebraic rank, respectively, by

m = ord_(s=1) L(E,s)

r = rank E(Q).

The BSD rank conjecture asserts that

For every elliptic curve E/Q, ord_(s=1) L(E,s) = rank E(Q).

For analytic ranks zero and one, the paper presents the established results in BSD theory.

Case m = 0:

For every E/Q, m = 0 implies r = 0.

Case m = 1:

For every E/Q, m = 1 implies r = 1.

For the general construction, the paper introduces two complex vector spaces associated with the elliptic curve:

A_E = C{t} / (L(E,1+t))

V_E = E(Q) tensor_Z C.

Their dimensions are

dim_C A_E = m, dim_C V_E = r.

The paper examines the symmetry operations

tau([f(t)]) = [f(-t)]

iota(v) = -v

and considers whether a complex-linear isomorphism

Phi: A_E -> V_E

can satisfy the symmetry-compatibility condition

Phi o tau = iota o Phi.

The properties of these operations show that no such symmetry-compatible isomorphism exists when

m >= 1.

Thus, the paper describes the rank cases of the Birch-Swinnerton-Dyer conjecture that have been proved for elliptic curves over Q with analytic rank zero or one.

Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics
Truth Vector
Zenodo18 Oct 2026PreprintCC BY 4.0Full textZenodo
Structural Coherence and the Birch and Swinnerton-Dyer Conjecture - Coerência Estrutural e a Conjectura de Birch e Swinnerton-Dyer
Deni da Silva Saez
Zenodo22 Oct 2026PreprintCC BY 4.0Full textZenodo
The Absolute Spectral Resolution of the Birch and Swinnerton–Dyer Conjecture: Colombeau–Takesaki L-Operators, Galois-Equivariant Dimensional Isomorphism, and Selmer–Sobolev Finiteness of Ш(E/ℚ) within the ZFC Axiomatic Bridge
Seonggil Lee
Zenodo21 Oct 2026PreprintCC BY 4.0Full textZenodo
Spectral Geometry of Flytrons and the Birch–Swinnerton-Dyer Conjecture: A ZEBTS Framework
Anatolii Mukha
Zenodo24 Oct 2026PreprintCC BY 4.0Full textZenodo
Hecke Transfer Operators, Canonical Height Coercivity, and the Birch and Swinnerton-Dyer Formulas
Richard Santiago Zapata
Zenodo16 Oct 2026PreprintCC BY 4.0Full textZenodo

Cite

BibTeX RIS

BibTeX

@misc{tran2026birchswinnertondyer,
  title = {{The Birch–Swinnerton–Dyer Rank Problem: Reduction of the Analytic Rank Two Case}},
  author = {Son Tuyet Tran},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23491736},
  url = {https://zenodo.org/records/23491736},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The Birch–Swinnerton–Dyer Rank Problem: Reduction of the Analytic Rank Two Case
AU  - Son Tuyet Tran
PY  - 2026
DA  - 2026/10/20
PB  - Zenodo
DO  - 10.5281/zenodo.23491736
UR  - https://zenodo.org/records/23491736
AB  - This paper studies the Birch-Swinnerton-Dyer (BSD) conjecture for elliptic curves E defined over the rational number field Q. The central focus is the construction of a general structure arising from elliptic curves, followed by transformations that lead to a symmetry comparison between the analytic and algebraic sides. The results are established within the scope of the constructed structure and the specified symmetry-compatibility condition.  Define the analytic rank and algebraic rank, respectively, by  m = ord_(s=1) L(E,s)  r = rank E(Q).  The BSD rank conjecture asserts that  For every elliptic curve E/Q, ord_(s=1) L(E,s) = rank E(Q).  For analytic ranks zero and one, the paper presents the established results in BSD theory.  Case m = 0:  For every E/Q, m = 0 implies r = 0.  Case m = 1:  For every E/Q, m = 1 implies r = 1.  For the general construction, the paper introduces two complex vector spaces associated with the elliptic curve:  A_E = C{t} / (L(E,1+t))  V_E = E(Q) tensor_Z C.  Their dimensions are  dim_C A_E = m, dim_C V_E = r.  The paper examines the symmetry operations  tau([f(t)]) = [f(-t)]  iota(v) = -v  and considers whether a complex-linear isomorphism  Phi: A_E -> V_E  can satisfy the symmetry-compatibility condition  Phi o tau = iota o Phi.  The properties of these operations show that no such symmetry-compatible isomorphism exists when  m >= 1.  Thus, the paper describes the rank cases of the Birch-Swinnerton-Dyer conjecture that have been proved for elliptic curves over Q with analytic rank zero or one.
ER  -

Abstract

Related

Cite