16 problems

Stark–Heegner cycle rationality conjecture. These local classes should be the restrictions at $p$ of global cohomology classes in the semistable Bloch–Kato Selmer group defined ove…

Let $M/K$ be a rank-one motive with coefficient ring $R$, let $T$ be its $p$-adic realization, and put $r=r(M/K,R)$. Assume that the $\mathcal{R}_p$-module…

Local Bloch–Kato conjecture. The element

De Clercq–Florence conjecture. Surjectivity of the norm residue morphism follows from the fact that $\mathcal{G}_{\mathbb{K},p}$ is 1-cyclotomic.

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

Let $N>1$ be a positive integer, let $\theta:(\mathbb{Z}/N\mathbb{Z})^\times\to\overline{\mathbb{Q}}^\times$ be an even Dirichlet character, and let $f\in S_2(N,\theta)$ be a norma…

Let $f_1,f_2,f_3$ be the triple of modular forms in the paper, let $\mathrm{V}(\underline{\mathbf{f}})$ be the associated $l$-adic representation over $E_{\underline{\lambda}}$, an…

Let $K$ be the relevant number field, let $f$ be the Hilbert modular form under consideration, and let $A(f)$ be its associated Galois representation. Write $Sel(K,A(f))$ for its S…

Let $E/\mathbb{Q}$ be the elliptic curve and let $F/\mathbb{Q}$ be the fixed totally real extension of odd degree. Assume that $\Sha(E/F)[p^\infty]$ is finite. Let $T$ be the assoc…

Let $\rho^\theta_\blambda$ be the $p$-adic realization of the motive associated with the Siegel modular form, let $rho^thetablambda aturaltheta$ denote its twist by $theta$, and le…

Let $g$ be a normalised cuspidal Hilbert modular newform of weight $(k_1,k_2,\ldots,k_d,w=2)$ and level $\Gamma_0(\mathfrak{n})$ over a totally real number field $F$, with each…

Let $(E,A)$ be an arbitrary pair, with associated Chow motive $\mathsf{M}_{E,A}$, Chow group $\operatorname{CH}(\mathsf{M}_{E,A})$, Bloch–Kato Selmer groups…

Nonvanishing conjecture. These equivalent properties always hold; in particular, $L_p(\psi^{-1}\tau,z^{k+1})\neq 0$. This conjecture concerns the expected multiplicity one for the…

Integral $d\log$ conjecture. The map $d\log_m$ is surjective. The rationalized map is the generic-point cycle class map $\lim(\textup{cl}_{m,m})$; the source notes that rational su…

Fontaine–Perrin-Riou extension of the Bloch–Kato conjecture. The following equality of fractional ideals of $O_K[1/S]$ holds:

Let $U$ be a smooth variety over a number field or a local field $F$, and let $\HM^i(U,n)$ be its motivic cohomology. Let $\HMf^i(U,n)$ be the subgroup whose $\ell$-adic Abel–Jacob…