Abstract An important conjecture in Iwasawa theory predicts that the fine Selmer group of any p -adic Galois representation is cotorsion over the relevant Iwasawa algebra with vanishing μ $\mu $ mu -invariant, generalizing Iwasawa’s original conjecture for class groups. In this article, we recast this conjecture in terms of higher Galois cohomological operations called Massey products, stated purely in terms of the residual representation. This characterization implies that if the μ $\mu $ mu -invariant vanishes for a given Z p $\mathbb {Z}_p$ double struck upper Z Subscript p -extension, then it also vanishes for all Z p $\mathbb {Z}_p$ double struck upper Z Subscript p -extensions the Greenberg neighborhood of radius 1 / p $1/p$ 1 divided by p . Furthermore, we establish that the unobstructedness of deformation rings over Z p $\mathbb {Z}_p$ double struck upper Z Subscript p -extensions is equivalent to the vanishing of the μ $\mu $ mu -invariant of the fine Selmer group attached to the adjoint representation. For ordinary deformation rings attached to two-dimensional automorphic Galois representations, we establish a close connection between the Noetherian property and the vanishing of the μ $\mu $ mu -invariant.
Abstract This article investigates how classical homological invariants extend from module categories to a triangulated category T $\mathcal {T}$ script upper T with a compact silting generator M . We construct sup-projective, inf-injective, and sup-flat resolutions for objects in T $\mathcal {T}$ script upper T and use them to establish basic properties of the corresponding homological dimensions. We then introduce the global dimension gldim T $\mathrm {gldim}\mathcal {T}$ gldim script upper T and prove that it coincides with the global dimension of the heart H M = M ⊥ ≠ 0 $\mathcal {H}_M=M^{\bot _{\neq 0}}$ script upper H Subscript upper M Baseline equals upper M Superscript normal up tack Super Subscript not equals 0 and with the Add ( M ) $\mathrm {Add}(M)$ upper A d d left parenthesis upper M right parenthesis -dimension of T b $\mathcal {T}^{\mathrm {b}}$ script upper T Superscript normal b when M is tilting. These results provide a unified framework for homological dimension theory in triangulated categories and reveal how silting and tilting objects encode essential homological information. As applications, we study homological properties, such as Gorensteinness and regularity for T $\mathcal {T}$ script upper T with a focus on the derived category over rings and dg rings.
Abstract This article investigates coefficient randomizations of holomorphic functions and the resulting random analytic functions within the framework of mixed norm spaces. Adopting the perspective of the random symbol space ( X ) ⋆ = { f ∈ H ( B n ) : P ( R f ∈ X ) = 1 } $(\mathcal {X})_{\star }=\{f\in H(\mathbb {B}_n): \mathbb {P}(\mathcal {R} f\in \mathcal {X})=1\}$ left parenthesis script upper X right parenthesis Subscript star Baseline equals StartSet f element of upper H left parenthesis double struck upper B Subscript n Baseline right parenthesis colon double struck upper P left parenthesis script upper R f element of script upper X right parenthesis equals 1 EndSet , we first focus on the unit disk D $\mathbb {D}$ double struck upper D . Specifically, for q ≥ 4 $q\geq 4$ q greater than or equals 4 , we identify the conditions under which the random symbol space of the mixed-norm space H p , q , u ( D ) $H^{p,q,u}(\mathbb {D})$ upper H Superscript p comma q comma u Baseline left parenthesis double struck upper D right parenthesis coincides with H 2 , q , u ( D ) $H^{2,q,u}(\mathbb {D})$ upper H Superscript 2 comma q comma u Baseline left parenthesis double struck upper D right parenthesis . Furthermore, we extend our analysis to the higher-dimensional unit ball B n $\mathbb {B}_n$ double struck upper B Subscript n . We establish sufficient conditions and necessary constraints linking the deterministic membership of a function to the almost sure membership of its randomization in mixed norm spaces. A key finding is the identification of a sharp, dimension-dependent weight loss phenomenon that arises for n > 1 $n>1$ n greater than 1 .
Abstract In this article, we study Iwasawa theory for the Selmer group of an elliptic curve over a certain non-commutative p -adic Lie extension whose Galois group is isomorphic to an open subgroup of PGL 2 ( Z p ) $\mathrm {PGL}_2(\mathbb {Z}_p)$ upper P upper G upper L 2 left parenthesis double struck upper Z Subscript p Baseline right parenthesis . Unlike the classes of extension usually considered in non-commutative Iwasawa theory, this extension does not contain the cyclotomic Z p $\mathbb {Z}_p$ double struck upper Z Subscript p -extension. We give a formula for its Euler characteristic in this case. We also show that the dual Selmer group contains no nonzero pseudonull submodules, a result analogous to a theorem of Ochi and Venjakob for GL 2 $\mathrm {GL}_2$ upper G upper L 2 -extensions. Finally, we provide an explicit formula computing the change in Iwasawa μ $\mu $ mu -invariants of dual Selmer groups when two elliptic curves are isogenous; this in the same spirit as that of Howson’s result for the GL 2 $\mathrm {GL}_2$ upper G upper L 2 -extension which extends results in the commutative case by Schneider and Perrin-Riou.
