25 problems

Let $f$ be the modular form, $F$ the field, $p$ and $k$ the prime and weight from Theorem 5.11, and let $u_2(f_F)$, $\Omega_f^+$, $\Omega_f^-$, and $R_{f,\alpha}$ be respectively t…

For each prime $p$, let $V_p:=V_{f,\mathfrak{p}}$ be the $p$-adic Galois representation attached to a newform $f$ of weight $2k-2$, let $T_p:=T_{f,\mathfrak{p}}$ be a stable lattic…

Equivariant Bloch–Kato conjecture. The Beilinson regulator induces an isomorphism $\Delta_f(G,M^\lor(1-k))\otimes {\Bbb{R}}\cong \1_{{\Bbb{R}}[G]}$, under which…

Let $M$ be a critical object of $\operatorname{{\bf PM}}_K$, let $b(M)$ be the basis supplied by Deligne's period conjecture, let $\lambda$ be a place of $K$ for which the stated v…

Let $M$ be an $L$-admissible premotivic structure over $K$, let $\tau:K\to\mathbb{C}$ be an embedding, and let $\lambda$ be any finite prime of $K$. Write…

Let $F$ be a number field, let $L$ be a $p$-adic local field, and let $V$ and $V'$ be geometric symplectic self-dual $p$-adic representations of $G_F$ over $L$. For each representa…

Let $F$ be a number field, let $L$ be a $p$-adic local field, and let $V$ be a geometric $p$-adic representation of $G_F$ over $L$, meaning that it is unramified outside finitely m…

Let $p$ be a prime, let $cK$ be a $p$-adic field, and let $V$ be a geometric irreducible representation of $\mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q)$ over $\mathcal K$. Write…

Let $X$ be as above, let $p$ be a prime of good reduction, and let $m<2r-1$. Write $H^m_{\operatorname{\acute{e}t}}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_p(r))$ for the $p$-adic éta…

Weight $k-2$ p-newness conjecture. The class $c^{”}$ is $p$-new for every $p\in\mathcal{S}$. This is motivated by the expectation that the general newform Eisenstein congruence con…

Cohomological p-newness conjecture. The class $c'$ is $p$-new for every $p\in\mathcal{S}$, meaning that it does not come from the corresponding relaxed Selmer group with the condit…

Bloch–Kato vanishing conjecture. One expects

Let $K$ be a number field and let $\text{Ш}^{\operatorname{BK}}(K,\mathcal{M})=\bigoplus_p\text{Ш}_p^{\operatorname{BK}}(K,\mathcal{M})$ be the Bloch–Kato Shafarevich–Tate group of…

Bloch–Kato conjecture. If $L(\pi,1+j)\ne 0$ for some integer $j$ with $t_i\leqslant j\leqslant k_i+t_i-2$ for all $i$, then

Rank-zero Bloch–Kato conjecture. Then

Let $\pi$ be an automorphic representation with associated Galois representation $\rho_\pi$, let $\chi$ be a character, and let $\mathcal{K}$ be the relevant number field. Write…

Let $M$ be a motive with $L$-function $L(M,s)$ and $p$-adic Selmer group $\mathrm{Sel}_{p^\infty}(M)$. Suppose its functional equation is … with $s=\frac{1}{2}$ a critical value. B…

Let $\mathcal{G}=(G,\theta)$ be a torsion-free 1-smooth pro-$p$ pair. Pro-$p$ smoothness conjecture. The pair $\mathcal{G}$ is weakly Bloch--Kato. This is the pro-$p$ version of th…

Smoothness Conjecture. If $G$ is $1$-smooth, then it has the weak Bloch–Kato property. In particular, it is smooth.

Let $\mathcal{M}=\mathcal{M}(\pi,r^{\pm}\otimes\Psi)$ be critical, with Betti and de Rham realizations $H_{\rm B}(\mathcal{M})$ and $H_{\rm dR}(\mathcal{M})$, and let…

Let $f$ be a newform of even weight $k$, let $0\leq j\leq k-2$, and assume $L(f,j+1)\neq0$. Let $\Omega^{(-1)^{j+1}}$ denote the relevant period, $u_{j+1}$ the associated unit fact…

Let $K/\mathbf{Q}$ be the quadratic extension, let $\pi$ be the automorphic representation and let $q>2k$ be a prime at which neither $\pi$ nor $K/\mathbf{Q}$ is ramified. Let…

Let $F$ be an imaginary quadratic field, let $\pi$ be a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_n$ over $F$ satisfying…

Bloch–Kato submodule conjecture. There is an $R$-submodule, called the Bloch–Kato submodule,

Let $G={\mathfrak G}_{\ell}$ be an $\ell$-Sylow subgroup of the absolute Galois group of a function field over an algebraically closed field. Set … where $G^{(1)}=[G,G]$ and…