38 problems

Let $u$ be a positive integer congruent to $4$ modulo $8$, such that $\operatorname{ord}_p(u(u^2-1))$ is zero or odd for every odd prime $p$. Assume conditions (1), (3), and (4) of…

Let $E/F$ be a CM extension, let $\Pi_n$ and $\Pi_{n+1}$ be RACSDC automorphic representations as above, and set $\Pi=\Pi_n\boxtimes\Pi_{n+1}$. Assume that their weights are perfec…

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…

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…

Beilinson–Deligne conjecture. The order of vanishing of $\operatorname{L}(\operatorname{Ad}^0(\rho_{f,p}),s)$ at $s=1$ equals

Compatibility conjecture. The element $\widetilde{\delta}_p(G_\infty,M^{\mathrm{triv}},k)$ should induce $\delta_p(G_n,M,k)$ for all $n$.

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 $E_u$ be the elliptic curve under consideration, let $R_u$ be the regulator obtained from the integral motivic lattice, and for each finite prime $\ell$ let $T_\ell$ and…

Let $u$ be as in Theorem 1. Assume that $\operatorname{INT} E_u\otimes_{\mathbb{Z}}\mathbb{Q}$ is one-dimensional and that … Let $\iota_{u,2}$ be the index defined after Theorem 1,…

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 $X$ be a smooth projective variety over a number field, let $i$ be an integer, and consider the cycle-class map … Here $\mathrm{CH}^i(X)_0$ denotes homologically trivial codime…

Refined Bloch–Kato conjecture. The following equality of fractional $\mathcal{O}_p$-ideals holds:

Normalized $L$-value conjecture. One has

Let $V(j)$ be the $j$th Tate twist of the relevant modular Galois representation in the critical range, and let $H^1_{\mathrm f}(V(j))$ and $H^1_{\{p\}}(V(j))$ denote the Bloch--Ka…

Let $V$ be a $G_{\Sigma}$-module valued in a coefficient field $E$, with ring of integers $\mathcal{O}$ and uniformizer $\varpi$. Let…

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…

Let $F$ be a number field and let $\rho$ be a Galois representation unramified outside a finite set of places and de Rham at every place above $p$. Write $\ad\rho$ for its adjoint…

Let $f_{\mathrm{new}} \in S_k(\Gamma_0(M))$ be a classical cuspidal newform with $k\geq 2$ and $p\nmid M$. Its analytic $L$-function is expected to encode arithmetic invariants of…

Rank-zero Bloch–Kato conjecture. Then

Rationality conjecture. The value $L(\operatorname{ad}^0(M),1)$, after normalization by some suitable period, should lie in $\mathbf{Q}^{\times}$.