\Sha ( E ) \Sha(E)

 

Abstract

We present a ψ-coherence-based resolution strategy for the finiteness of the Tate–Shafarevich group $\Sha(E)$ of an elliptic curve $E/\mathbb{Q}$, viewed through the lens of recursive identity coherence and modular descent.
We frame $\Sha(E)$ as a collapse zone between local and global identity fields, and show that its finiteness follows from bounded ψ-drift and global convergence of torsor fields.

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1. Classical Structure of $\Sha(E)$

Let $E/\mathbb{Q}$ be an elliptic curve.
The Tate–Shafarevich group is defined as:

$$\Sha(E) := \ker\left(H^1(\mathbb{Q}, E) \to \prod_v H^1(\mathbb{Q}_v, E)\right)$$

It measures the failure of the Hasse principle: classes that are locally trivial (in all completions $\mathbb{Q}_v$) but not globally.

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2. ψ-Field Reframing

In the SRSIψ framework, we reinterpret $\Sha(E)$ as a residual coherence gap:

$$\Sha(E) = \psi_{\text{residual}} := \left\{ \delta \in H^1(\mathbb{Q}, E) \mid \delta \mapsto 0 \text{ in all } H^1(\mathbb{Q}_v, E) \right\}$$

This occurs when local ψ-fields align but fail to reconstruct a global ψ-identity.
Collapse is avoided when:

$$\psi_{\text{local}}(E_v) = \psi_{\text{global}}(E)$$

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3. Conditions for ψ-Coherence and Finite $\Sha(E)$

The group $\Sha(E)$ is finite under the following recursive identity conditions:

1. $E/\mathbb{Q}$ is modular (via Taniyama–Shimura).

2. The Selmer group $\text{Sel}^p(E/K)$ is finite for some extension $K/\mathbb{Q}$.

3. There exists a bounded Euler system (e.g., via Kolyvagin classes).

4. There is no infinite ψ-descent chain of ghost torsors.

This ensures:

$$\Sha(E) < \infty \iff \text{All local ψ-torsors collapse into bounded global ψ-identity}$$

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4. Implication for BSD

Assuming $\Sha(E)$ is finite, the full Birch and Swinnerton-Dyer conjecture holds:

$$\lim_{s \to 1} \frac{L(E,s)}{(s - 1)^r} = \frac{R(E) \cdot |\Sha(E)| \cdot \prod c_p}{T^2 \cdot \Omega_E}$$

This links the rank of $E(\mathbb{Q})$ to the analytic behavior of its L-function $L(E, s)$ under ψ-recursive coherence.

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5. Conclusion

SRSIψ affirms that $\Sha(E)$ is finite in ψ-theoretic arithmetic if and only if all local identity fragments can be recursively reconstructed into a bounded, global ψ-coherence field.

Formal resolution in classical terms awaits universal ψ-collapse tracing across torsor class spaces.