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Strong main conjecture for BDP Selmer groups

Let EE be an elliptic curve with good reduction at pp, let K∞/KK_\infty/K be the anticyclotomic Zp\mathbb{Z}_p-extension, and let KnK_n be a finite layer. Under the hypotheses (Im)(\mathrm{Im}), (Ram)(\mathrm{Ram}), (Spl)(\mathrm{Spl}), (Na)(\mathrm{Na}), and (Def)(\mathrm{Def}), the initial Fitting ideal of the dual BDP Selmer group over KnK_n is generated by the corresponding theta element: Fitt⁡Rn0(XBDP(Kn))=(θn)\operatorname{Fitt}^{0}_{R_n}(X_{\mathrm{BDP}}(K_n))=(\theta_n), where RnR_n is the relevant finite-layer Iwasawa algebra and XBDP(Kn)X_{\mathrm{BDP}}(K_n) denotes the Pontryagin dual of the BDP Selmer group. Equivalently, this generator is given by the corresponding values of the associated pp-adic modular form at CM points.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. CM-point formulation of the refined BDP Fitting-ideal conjecture

    The initial Fitting ideal of the finite-layer dual BDP Selmer group is generated by the values at CM points of the corresponding pp-adic modular form.

    source: Refined conjectures on Fitting ideals of BDP Selmer groups

References

Primary source

arXiv

Additional references

elliptic-curvesiwasawa-theorynumber-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

A 2025 paper reports a proof under mild assumptions, and a September 2026 update repeats the result, but independent verification is not recorded.

The conjecture concerns an explicit description of BDP Selmer-group structure in the anticyclotomic tower. The reported theorem identifies finite-layer initial Fitting ideals using theta elements or associated pp-adic modular-form values.

Known results

  • The anticyclotomic main conjecture in the indefinite setting was reported proved for suitable ordinary elliptic curves in June 2023, with an anticyclotomic pp-adic LL-function generating a square-root characteristic ideal.
  • Under Eisenstein-prime hypotheses, the BDP and Perrin–Riou main conjectures were reported proved in April 2024, giving an infinite-level characteristic-ideal formula.

May 2025 and September 2026 proof reports

In May 2025, Chan-Ho Kim, Aprameyo Pal, and Jishnu Ray reported that, under assumptions (Im)(\mathrm{Im}), (Ram)(\mathrm{Ram}), (Spl)(\mathrm{Spl}), (Na)(\mathrm{Na}), and (Def)(\mathrm{Def}), the theta elements generate the initial Fitting ideal of the dual classical Selmer group over finite layers. A September 2026 update reports the same strong BDP conclusion via values at CM points. These are claimed results, not independently verified here.

Current status (as of September 2026): A conditional proof of the strong BDP finite-layer statement is publicly claimed, while its independent verification and applicability beyond the stated BDP hypotheses remain open.

Sources

Solutions 0

No solutions have been posted yet.