Sea f una forma modular nueva como en el contexto del texto, y sea X(f) el módulo de Selmer asociado, considerado como módulo de torsión sobre el álgebra de Iwasawa Lambda(G). Sea mu_f el elemento p-ádico asociado a la función L de f. Mazur's main conjecture for modular forms. operatornamecar_Lambda(G) X(f)=(mu_f). La afirmación extiende la conjetura principal para curvas elípticas al contexto de formas modulares; el caso de curvas elípticas aparece como caso especial. El texto no proporciona una resolución general.

Mazur's main conjecture for modular forms

The conjecture asks whether the characteristic ideal of the Selmer module equals the ideal generated by the associated $p$ -adic $L$ -function, extending the elliptic-curve case. No source found a general proof or counterexample.

Known results

$2016$ : An anticyclotomic analogue of the weak Mazur–Tate conjecture was proved under hypotheses.
$2019$ : Kato’s main conjecture was shown to propagate within suitable families, assuming it for one form and additional hypotheses.
$2021$ : A cyclotomic main conjecture for individual forms and universal families was claimed under residual-representation hypotheses.
$2022$ : A weak Mazur–Tate inclusion and finiteness statement was proved in a cyclotomic setting.

2026 anticyclotomic main-conjecture claims

A 2026 preprint claims proofs in definite and rank-one indefinite anticyclotomic settings, under arithmetic hypotheses. These are substantive special cases, but they do not establish the general equality in the problem.

Current status (as of August 2026): The general conjecture remains open; several conditional cyclotomic, family-level, and anticyclotomic cases are claimed or established, but none settles the stated formulation.

Mazur's main conjecture for modular forms

Primary source

Progress summary

Solutions 0