26 problems

Let $\bar{\rho}:G_{\mathbf{Q}}\to\operatorname{GL}_{2}(k)$ be the residual representation, let $\mathcal H(\bar{\rho})$ be its Hida family, and let $Y$ be a branch of this family.…

Big $P$-ordinary Hecke algebra conjecture. 1. For each tame character $\epsilon$, $\mathbf{T}^{P\text{-ord}}_{K^p,[\kappa,\tau],R,\epsilon}$ is finite free over…

Independence-of-weight conjecture. If $\kappa_1$ and $\kappa_2$ are dominant characters such that $[\kappa_1]=[\kappa_2]$, then there is a canonical isomorphism

Let $\bT_{\mathfrak{m}}$ be the relevant ordinary Hecke algebra, let $\mathcal{I}_{\mathfrak{m}}$ be its congruence ideal, and let $L_p^+(\mathfrak{m},\omega^j)$ be the associated…

Let $\mathcal{I}_{\mathfrak{m}}$ be generated by $\mathfrak{t}$. For every height-one prime $\mathfrak{p} \subset \bT_{\mathfrak{m}}^0$, let $\mathcal{O}_{\mathfrak{p}}$ be the nor…

Retain the Hida-family settings of the source. Let $\mathcal{A}$ be the big Galois representation, let $F_\infty$ be a $\mathbb{Z}_{p}$-extension of a number field $F$, and write…

Let $\mathbb{I}$ be a UFD and Gorenstein, assume $\chi|_{((\mathbb{Z}/p\mathbb{Z})^{\times})}\neq\mathbf{1},\omega$, $p\nmid\varphi(N)$, and that the Eisenstein ideal $\mathbb{J}$…

Let $d53c$ be the coefficient ring of a residually reducible Hida deformation, let $d54b$ be a stable free lattice, and put … Write $\operatorname{Sel}_{\mathcal{A}}$ for the assoc…

Let $(\Pi,\underline{\mathcal G})$ be a split pair with $\Pi_p$ ordinary, and let $\mathfrak X_\spadesuit$ be the classical specialisations in a region $\spadesuit$ whose archimede…

Let $\operatorname{\boldsymbol f}^{(p)}$ be the $p$-ordinary Hida family associated with a modular form, and let $f_\wp^\flat$ be the newform attached to its arithmetic specializat…

Assume that $\mathbb{I}=\mathfrak{O}_L[[T]]$, and for each $k\in\mathbf{Z}_{\geqslant 2}$ let $\boldsymbol{f}_k$ be the $p$-stabilized newform obtained by setting $T=(1+p)^{k-2}-1$…

Let $\mathcal{R}$ be a primitive branch of a Hida family, let $\tilde{\mathcal{X}}(\mathcal{R})$ be the metaplectic covering of its arithmetic weight space, and let…

Let $\mathcal{R}$ be a primitive branch of a Hida family. Its Big Heegner points determine proportionality coefficients in the fraction field of $\mathcal{R}$, and let $\mathbb{S}$…

Let $\mathcal E_{\mathrm{par}}$ be the parallel-weight eigenvariety, let $\mathcal C_M$ denote the relevant Coleman–Mazur eigencurve at level parameter $M$, and let $[\varphi]$ den…

Let $\mathbf{f}$ be a Hida family of $p$-ordinary $p$-stabilized cusp newforms of tame level $N$, and let $\mathcal{T}$ be the nearly ordinary Hida deformation associated with…

Main conjecture over $\mathbb I[[{\mathcal G}]]$. There exists an element $\widetilde\xi\in K'_1(\mathbb I[[{\mathcal G}]]_{\mathscr S})$ such that

Let $L_p(s,\kappa)$ be the two-variable $p$-adic $L$-function attached to a Hida family, with $s$ the $p$-adic $L$-function variable and $\kappa$ the weight variable. Consider the…

Let $L_p(s,\kappa)$ be the two-variable $p$-adic $L$-function attached to a Hida family, with $s$ the $p$-adic $L$-function variable and $\kappa$ the weight variable. Consider the…

To a modular form $g$, Hida attaches a $q$-expansion $\mathbf{g}\in R[[q]]$, where $R$ is an integral domain finite and flat over the Iwasawa algebra…

Horizontal nonvanishing conjecture. The cohomology class $\mathfrak{Z}_0$ is not $R$-torsion.

The two-variable Heegner point main conjecture. Assuming that $R$ is regular,

Let $K$ be the imaginary quadratic field, let $\mathcal R$ be the Hida-family coefficient ring, and let $\mathfrak Z_0\in\widetilde H^1_f(K,\mathbf T^\dagger)$ be the corestriction…

Let $f_\mathfrak p^\dagger$ be a non-exceptional arithmetic specialization of weight $k_\mathfrak p\ge2$, let $\chi$ be a finite-order character of $G_\infty$, and let…

Let $f_\mathfrak p^\dagger$ be the twisted specialization at an arithmetic prime $\mathfrak p$ of weight $k_\mathfrak p\ge2$, let $\chi$ be a fixed character of $\widetilde G_c$ as…

Let $\mathcal R$ be the local Hida-family coefficient ring, assumed regular, and let $\mathcal R_\infty$ be the coefficient ring extended by the anticyclotomic $\mathbb Z_p$-extens…