[Paper Review] An anticyclotomic Mazur-Tate conjecture for modular forms

Chan-Ho Kim|arXiv (Cornell University)|Dec 12, 2016

Algebraic Geometry and Number Theory29 references3 citations

TL;DR

This paper proves an anticyclotomic Mazur-Tate conjecture for modular forms over cyclic ring class extensions of imaginary quadratic fields, extending the Euler system divisibility from p-adic Z_p-extensions to more general finite abelian extensions. By generalizing the Euler system argument to include prime-to-p extensions and establishing a refined control theorem, the author derives Fitting ideal bounds on Selmer groups, providing a structural refinement of the Iwasawa main conjecture beyond mere size estimates.

ABSTRACT

Under certain assumptions, we prove an anticyclotomic analogue of the "weak main conjecture" à la Mazur and Tate for modular forms over a large class of cyclic ring class extensions.

Motivation & Objective

• To extend the Mazur-Tate conjecture from p-adic Z_p-extensions to general cyclic ring class extensions of imaginary quadratic fields.
• To refine the Iwasawa main conjecture by studying Fitting ideals over finite abelian extensions rather than just characteristic ideals over the infinite Z_p-extension.
• To establish a generalized Euler system divisibility for the anticyclotic Iwasawa main conjecture over extensions with Galois group Z_p × Z/m'Z, (p, m')=1.
• To analyze the structure of Selmer groups via Fitting ideals, offering a deeper arithmetic understanding beyond the size of these groups.

Proposed method

• Generalizes the Euler system argument of Bertolini-Darmon to include prime-to-p cyclic ring class extensions by upgrading the Chebotarev density result to more general base fields.
• Applies a character-by-character divisibility argument over discrete valuation rings, specializing via characters to reduce the problem to local computations.
• Develops a generalized lifting argument to lift divisibilities from discrete valuation rings to the generalized Iwasawa algebra, overcoming limitations of standard Nakayama-type arguments in semi-local rings.
• Uses isotypic decomposition with respect to quotients to recycle the original Euler system strategy and adapt it to non-p-primary extensions.
• Establishes a control theorem for compact Selmer groups with respect to quotients, relying on a generalized freeness theorem for Selmer groups over the extended base fields.
• Introduces a mod p^n level-raising construction at n-admissible primes and emphasizes the role of 'Condition Ihara' in the non-ordinary setting.

Experimental results

Research questions

• RQ1 Can the Mazur-Tate conjecture be extended from p-adic Z_p-extensions to more general cyclic ring class extensions of imaginary quadratic fields?
• RQ2 How can the Euler system argument be generalized to work over extensions with Galois group Z_p × Z/m'Z, (p, m')=1, rather than just p-power extensions?
• RQ3 What is the role of the generalized Iwasawa algebra and how can one lift divisibility results from discrete valuation rings to this algebra in the semi-local setting?
• RQ4 How does the structure of Selmer groups change when studying Fitting ideals over finite abelian extensions instead of characteristic ideals over the infinite Z_p-extension?
• RQ5 What is the significance of 'Condition Ihara' in the context of level-raising and exceptional zero phenomena for non-p-stabilized modular forms?

Key findings

• The paper establishes the Euler system divisibility of the generalized anticyclotomic Iwasawa main conjecture over extensions with Galois group Z_p × Z/m'Z, (p, m')=1, under suitable assumptions.
• It proves that the generalized Bertolini-Darmon element lies in the Fitting ideal of the dual Selmer group over cyclic ring class extensions, i.e., $ L_M(K(m),E) \in \mathrm{Fitt}_{\mathbb{Z}/M\mathbb{Z}[\Gamma_m]}(\mathrm{Sel}(K(m),E[M])^\vee) $.
• For each prime power $ p^{r_i} $ dividing $ M $, the divisibility $ L_M(K(m),f) \mod p^{r_i} \in \mathrm{Fitt}_{\mathbb{Z}/p^{r_i}\mathbb{Z}[\Gamma_m]}(\mathrm{Sel}(K(m),E[p^{r_i}])^\vee) $ holds, enabling a global bound via the Chinese Remainder Theorem.
• The method reveals a new type of exceptional zero phenomenon arising from prime-to-p extensions, distinct from the classical p-adic exceptional zero case.
• The generalized control theorem and freeness theorem for compact Selmer groups are extended to non-p-primary extensions using n-admissible sets and isotypic decomposition.
• Corollary 17.4 provides a partial answer to the size of the Shafarevich-Tate group over cyclic ring class extensions for composite $ M $, giving a bound on the $ \chi $-isotypic part in terms of the $ p $-adic valuation of the generalized L-invariant.