19 problems

Schneider's nonvanishing conjecture. The Schneider regulator with respect to the cyclotomic character is nonzero. This conjecture asserts nondegeneracy of the cyclotomic $p$-adic h…

Let $F_\infty$ be the cyclotomic $\mathbf Z_p$-extension of $F$, and let $h^{(1)}$ denote the first derived height pairing. Cyclotomic nondegeneracy conjecture. The pairing…

Non-degeneracy conjecture. When $r=0$, the pairing $[\,,\,]_{K,{\mathfrak p}^*}$ is non-degenerate.

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

Height Conjecture. The homomorphism $\mathrm{h}$ is an isomorphism of free $\Lambda_{\mathrm{anti}}$-modules of rank one.

Unit-root conjecture. $W^{C}_v$ is the unit-root subspace when the reduction of $C$ at $v$ is semistable ordinary.

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…

Let $F$ be a number field, let $v$ be a finite place above a prime $p\geq 3$, and let $J/F$ be a Jacobian with semistable ordinary reduction at $v$. Let $\mathcal{J}$ be the Néron…

Mazur–Bertolini–Darmon conjecture. The dimensions satisfy

Assume Hypotheses $ref{hy:galois}$ and $ref{hy:function}$. Let $\mathscr{L}^1_{\mathcal{E}_{mathtt{P}}}(Pi)$ be the height-valued $p$-adic $L$-function and let…

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.…

Let $E$ be an elliptic curve over a number field $F$ with good ordinary reduction at $p$. The associated $p$-adic height pairing on $E(F)$ is a bilinear pairing whose determinant i…

Assume that $E$ has either good ordinary or multiplicative reduction at $p$. The associated $p$-adic height pairing is a pairing on the relevant Mordell–Weil group, and the $p$-adi…

Let $A$ be an abelian variety over a number field $F$, let $A^{\vee}$ be its dual, and suppose that for every $v\mid p$ the representation…

Bertolini–Darmon–Mazur conjecture. In this situation,

Derived-height structure conjecture. There should be a $\Lambda$-module $M'$ with characteristic ideal prime to $J$ such that

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…

Non-degeneracy conjecture for the restricted p-adic height pairing. The pairing $[\,,\,]_{K,{\mathfrak p}^*}$ is non-degenerate when $r=0$. For positive rank, the paper notes non-d…