12 problems

Let $F$) be a totally real field, let $p$ be a prime, let $H_{\infty}/F$ be the relevant infinite extension with Galois group $\mathcal{G}$, and let $S$, $S_{ram}$, and $S_p$ denot…

Let $E$ be an elliptic curve with good ordinary reduction at an odd prime $p$, let $\theta_n(f)$ be the Mazur–Tate element attached to the modular form $f$ associated with $E$ at…

Let $H/F$ be a finite abelian CM extension, let $R=\mathbf{Z}[1/2][G]^-$, and define the Selmer module … with contragredient $G$-action. Let $\Theta_{S,T}$ be the smoothed Stickelb…

Let $R=\mathbf{Z}[1/2][G]^-$, let $\operatorname{Cl}^T(H)^-$ be the minus part of the $T$-smoothed class group, and for each ramified place $v$ let $A_v=\mathbf{Z}[G/I_v]/(g_v)$. W…

Let $H/F$ be a finite abelian CM-extension, and write $G=\operatorname{Gal}(H/F)$. Let $\Sigma$ and $\Sigma'$ be finite sets of places of $F$ satisfying (H1), (H2), (H3), and (H4).…

Let $F$ be the totally real field, let $S$ be the set of places dividing the level and $p$, and let $E/F$ be a totally real abelian extension ramified away from $S$ or at $p$. Set…

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…

Weak Fitting-ideal generation conjecture. The union of the sets

Minor ideal generation conjecture. For all $e\geq 0$, the ideal $\operatorname{Min}_e(A)$ is generated by $M(e,r-1-t_2+e)$.

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…

Let $K/k$ be a Galois extension of number fields with Galois group $G$, let $p$ be an odd prime, let $n>1$, and let $S$ and $T$ be finite non-empty disjoint sets of places of $k$,…

Let $S$ be a finite set of places of $k$ containing $S_\infty$ and all places ramified in $F/k$. Let $\lambda:Y_r\to X_r$ be a homomorphism with finite kernel and cokernel, let…