About the work
This paper establishes a new spectral–adèlic operator framework for the Hasse–Weil
$L$-functions of elliptic curves. Working entirely in the $\pi_E$-isotypic compression,
we construct explicit finite-rank operators $M_E(s)$ such that
\[
\det(I - M_E(s)) = c(s) L(E,s),
\]
with $c(s)$ holomorphic and non-vanishing at $s=1$.
From this central identity, we obtain:
- Unconditional proof of the Birch–Swinnerton–Dyer formula for analytic rank 0 and 1.
- A complete reduction of the conjecture in higher rank to two explicit compatibilities:
(dR) the local $p$-adic Hodge landing, and (PT) the spectral–Poitou–Tate pairing.
These compatibilities are proven in several key cases (good reduction, Steinberg,
supercuspidal depth 2, rank 1 via Gross–Zagier–Kolyvagin), and conjecturally equivalent
to the Beilinson–Bloch framework in general.
The analytic and spectral side of BSD is thus complete and unconditional; the arithmetic
side is reduced to finite, well-posed problems in $p$-adic Hodge theory and Selmer
cohomology.
In addition, we provide a computational implementation in SageMath/Python verifying
the framework across dozens of LMFDB curves, producing LaTeX finiteness certificates
for each curve. This makes the work fully reproducible and open to community validation.
We conclude that the Birch–Swinnerton–Dyer conjecture is no longer a global open problem
but a finite, explicit research program within reach of modern arithmetic geometry.
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Title A Spectral Resolution of the Birch–Swinnerton–Dyer Conjecture
This paper establishes a new spectral–adèlic operator framework for the Hasse–Weil
$L$-functions of elliptic curves. Working entirely in the $\pi_E$-isotypic compression,
we construct explicit finite-rank operators $M_E(s)$ such that
\[
\det(I - M_E(s)) = c(s) L(E,s),
\]
with $c(s)$ holomorphic and non-vanishing at $s=1$.
From this central identity, we obtain:
- Unconditional proof of the Birch–Swinnerton–Dyer formula for analytic rank 0 and 1.
- A complete reduction of the conjecture in higher rank to two explicit compatibilities:
(dR) the local $p$-adic Hodge landing, and (PT) the spectral–Poitou–Tate pairing.
These compatibilities are proven in several key cases (good reduction, Steinberg,
supercuspidal depth 2, rank 1 via Gross–Zagier–Kolyvagin), and conjecturally equivalent
to the Beilinson–Bloch framework in general.
The analytic and spectral side of BSD is thus complete and unconditional; the arithmetic
side is reduced to finite, well-posed problems in $p$-adic Hodge theory and Selmer
cohomology.
In addition, we provide a computational implementation in SageMath/Python verifying
the framework across dozens of LMFDB curves, producing LaTeX finiteness certificates
for each curve. This makes the work fully reproducible and open to community validation.
We conclude that the Birch–Swinnerton–Dyer conjecture is no longer a global open problem
but a finite, explicit research program within reach of modern arithmetic geometry.
Work type Education, Informative
Tags a spectral resolution of the birch–swinnerton–dye
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Identifier 2509303194051
Entry date Sep 30, 2025, 3:24 PM UTC
License All rights reserved
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Author 100.00 %. Holder JOSE MANUEL MOTA BURRUEZO. Date Sep 30, 2025.
Information available at https://www.safecreative.org/work/2509303194051-a-spectral-resolution-of-the-birch-swinnerton-dyer-conjecture
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Registration: 2509303194051
