7 problems

Let $V\in {\mathscr{V}}^{\circ}$ be an incoherent pair, and let $\pi$ be a distinguished, stable, ordinary, cuspidal automorphic representation of $\mathrm{G}^{V}(\mathbf{A})$, tri…

Maximal nondegeneracy conjecture. If $j$ is even and $0\leq j\leq 2\min(r^+,r^-)$, then

Let $F$ be a number field, let $v$ be a finite place above a prime $p$, and let $J/F$ be a Jacobian with semistable ordinary reduction at $v$. Let $\Theta$ be a theta divisor and s…

Assume that $Pi$ is incoherent and Hypothesis $ref{hy:galois}$ holds. Let $\mathcal{E}/E$ be a $mathtt{P}$-extension. Height-form main conjecture. The following are equivalent: (i)…

Assume Hypothesis $ref{hy:galois}$, let $\mathbb{V}_Pi$ be the associated Galois representation, and let $\mathscr{L}^1_\mathcal{E}(Pi)$ be the height-valued $p$-adic $L$-function.…

Bertolini–Darmon–Mazur conjecture. In this situation,

Let $A$ be an abelian variety over $F$ in the dihedral setting above $K$, let $G=\operatorname{Gal}(K_{p^\infty}/K)$, and write $\Lambda=\Lambda(G)={\bf Z}_p[[G]]$. Let…