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3 results

  • Kato’s main conjecture for potentially ordinary primes

  • Part of
  • Katharina Müller
  • Journal:
    Glasgow Mathematical Journal / Volume 66 / Issue 2 / May 2024
    Published online by Cambridge University Press:
    26 January 2024, pp. 338-358
    Print publication:
    May 2024
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  • In this paper, we prove Kato’s main conjecture for $CM$ modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.

  • On anticyclotomic μ-invariants of modular forms

  • Part of
  • Robert Pollack, Tom Weston
  • Journal:
    Compositio Mathematica / Volume 147 / Issue 5 / September 2011
    Published online by Cambridge University Press:
    27 July 2011, pp. 1353-1381
    Print publication:
    September 2011
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  • We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant.

  • THE ANTICYCLOTOMIC MAIN CONJECTURE FOR ELLIPTIC CURVES AT SUPERSINGULAR PRIMES

  • Henri Darmon, Adrian Iovita
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 7 / Issue 2 / April 2008
    Published online by Cambridge University Press:
    16 November 2007, pp. 291-325
    Print publication:
    April 2008

  • The Main Conjecture of Iwasawa theory for an elliptic curve $E$ over $\mathbb{Q}$ and the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ was studied in \cite{bertolini_darmon}, in the case where $p$ is a prime of ordinary reduction for $E$. Analogous results are formulated, and proved, in the case where $p$ is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou, Pollack, Kurihara, Kobayashi and Iovita and Pollack are required to handle this case in which many of the simplifying features of the ordinary setting break down.