41 problems

Let $U$ be a sufficiently small affinoid in the half-ordinary universal deformation family, let $V_U$ be its associated representation, and let $\mathscr{D}^+_U$ be the chosen Panc…

Let $\underline{\Pi}$ be a family on $\operatorname{GSp}_4$ with root number $-1$ and no endoscopic specialisations, let $\underline{\sigma}$ be the accompanying $\mathrm{GL}_2$ fa…

Let $\underline{\Pi}$ be a family on $\operatorname{GSp}_4$ whose specialisations have root number $-1$, let $\underline{\sigma}$ be the accompanying $\mathrm{GL}_2$ family, and le…

Let $k_\infty/k$ be a $\mathds Z_p$-power extension with no finite place splitting completely, let $K_\infty=Kk_\infty$, and use the source's decomposition and character $\chi$ to…

Let $\mathcal O$ be the coefficient ring, let $I$ be the relevant augmentation ideal, let $c\in\bigcap_{\Lambda_I}^r H^1_\Sigma(\mathcal O_{K,S},\mathbb T)_I$ be the special elemen…

Let $\underline{\pi}$ and $\underline{\sigma}$ be the Coleman families and let $\mathbf{z}_1(\underline{\pi}\times\underline{\sigma},\gamma_S)$ be the associated Euler-system cohom…

Euler system existence and nonvanishing conjecture. If $V$ is $r$-critical and satisfies the rank $r$ Panchishkin condition for $r\geq 0$, there exists a collection of cohomology c…

Let $T_f$ be a Galois-stable $\mathcal{O}_\pi$-lattice of $V_f$ satisfying Assumption (Im0), and let $r\in\mathbb{Z}$. Assume that $p$ does not divide $N$. For $i=0,\ldots,p-2$, le…

Let $K$ be a CM field, let $g=\tfrac12[K:\mathds{Q}]$, and fix a finite-prime-to-$p$ character $\chi:G_K\to\overline{\mathds{Q}_p}^{\times}$ satisfying the conditions in the source…

Let $L_\infty/K$ be the relevant $p$-adic extension, let $\Lambda=\mathds{Z}_p\llbracket\operatorname{Gal}(L_\infty/K)\rrbracket$, let $\varepsilon$ be the specified idempotent, an…

Soulé–Stark conjecture. The Soulé–Stark element coincides with the Bloch–Kato element:

Assume $\alpha_g\beta_h=1$, and let $P,Q$ be the points in $(E(H)\otimes V_{gh}^{\vee})^{G_{\mathbb Q}}$. Let $\kappa'(f,g_{\alpha},h_{1/\beta})$ be the derived cohomology class, l…

Let $F$ be the totally real field, let $S$ be the finite set of places occurring in the construction, and let $E/F$ be a totally real abelian extension ramified away from $S$ or at…

Let $K_\infty/k$ be the relevant $mathds{Z}_p$-extension, let $\Gamma$ be its pro-$p$ direction, let $W=V'\setminus V$ have cardinality $e$, and let…

Weak Leopoldt conjecture. For every $E\in\Omega(\mathcal{K}'/K)$, $H^2_{\mathrm{Iw}}(\mathcal{O}_{E,S(E)},T^*(1))$ is a torsion $\Lambda$-module.

Let $T_\Pi$ be the Galois lattice attached to $\Pi$, let $\mathcal O$ be its coefficient ring, and let $\mathcal N$ be the set of integers $mp^k$ where $m$ is a square-free product…

Let $\rho_x$ be a rank-two $p$-adic representation in the universal deformation family, let $\Sigma$ be a finite set of primes containing the ramified primes and $p$, and let…

Let $\rho_x$ be a rank-two $p$-adic representation in the universal deformation family, let $\Sigma$ be a finite set of primes containing the ramified primes and $p$, and let…

Let $\overline{\rho}$ be an absolutely irreducible residual rank-two representation, let $\Delta=\operatorname{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, and let…

The local Tamagawa number conjecture. For every positive integer $j$ there exists a unique $\mathbb{Z}_p$-basis

Assume the paper's Hypothesis, set $r=r_{\rm alg}$, and let $\bm{x}$ be a $\mathbb Z_p$-basis of $\bigwedge_{\mathbb Z_p}^{r-1}H^2(\mathbb Z_S,T)_{\rm tf}$. Let $\kappa_\infty$ be…

Let $F/\mathbb Q$ be the fixed finite abelian extension, let $G=\operatorname{Gal}(F/\mathbb Q)$, let $I_F$ be its augmentation ideal, let $T$ be the relevant $p$-adic representati…

Assume the paper's Hypothesis and set $r=r_{\rm alg}$. Let $F/\mathbb Q$ be the fixed finite real abelian extension, let $I_F$ be the augmentation ideal, let ${}_{c,d}z_F$ be the m…

Let $E/\mathbb Q$ be the elliptic curve, let $r_{\rm an}=\operatorname{ord}_{s=1}L(E,s)$ and $r_{\rm alg}=\operatorname{rank}E(\mathbb Q)$, and let ${}_{c,d}z_{\mathbb Q}$ be the…

Let $K$ be a number field, let $\mathcal{A}^{\rm s}_K$ be the specified ideal in the relevant group ring, let ${\rm ES}_K$ be the module of Euler systems of rank $r_K$ for…