22 problems

Let $K_S^{\mathrm{ab}}$ be the maximal abelian extension of $K$ unramified and totally split at the primes in $S$, let $\mathcal G=\operatorname{Gal}(K_S^{\mathrm{ab}}/K)$, and let…

Non-archimedean Darmon point conjecture. The local point $P_\psi$ is rational over $H_{\mathcal O}^+$, and, for every $\alpha\in\operatorname{Pic}^+({\mathcal O})$,

Tate lattice conjecture. The lattice $L$ is commensurable with the Tate lattice

Archimedean Darmon point conjecture. The isogeny $\beta$ can be chosen so that

Oda–Yoshida isogeny conjecture. There exists an isogeny

Darmon point globality conjecture. The isogeny $\beta$ can be chosen so that $P_\psi$ is the image of a global point in $E(H_{\mathcal O}^+)$, and, for every…

Isogeny conjecture. There exists an isogeny

Globality conjecture. These Darmon points should be global, namely defined over the corresponding ring class fields rather than only in the relevant local fields.

Let $K/F$ be a quadratic extension, let $\mathcal{O}$ be an order in $K$, let $K_{\mathcal{O}}^+$ be the associated narrow ring class field, and let $P_{q,b}$ be the Darmon point a…

Let $P_{q,b}\in E(\mathbb{C})$ be the Darmon point obtained by applying the Weierstrass uniformization to the period integral associated with the relevant automorphic form. Archime…

Let $K/F$ be a quadratic extension embedded in a quaternion algebra $B$, let $\mathcal{O}$ be an order in $K$, let $H^{\mathcal{O}}$ be its ring class field, and let $P_\tau$ be th…

Let $B$ be a quaternion algebra over a totally real field $F$, let $\Gamma$ be an Eichler order, let $K/F$ embed into $B$, and let $X$ be the associated symmetric space of dimensio…

Let $F$) be a totally real field, let $K/F$ be a quadratic extension with at least one real place, let $E/F$ be an elliptic curve, and let $\mathcal{O}\subseteq K$ be an…

Greenberg's rationality conjecture. The local point $P_\psi$ is a global point, and more precisely

Hilbert class field conjecture for ATR points. The ATR points are defined over the Hilbert class field of $K$.

Let $P_M^{\mathrm{Gar}}\in E(M)$ and $P_M\in E(M)$ be the two Darmon-type points constructed under the stated hypotheses, and let … Here $c_{E^\tau}$ is the real period or twice th…

Darmon-point conjectures. (1) $P_\varphi^\beta\in A_1(L_\varphi^\beta)$. (2) For every such $\chi$, the point $P_\chi$ is nonzero if and only if $L'(f_0/K,\chi,1)\ne0$. (3) For eve…

Yuan–Zhang–Zhang-type conjecture. There exists $b\in\widehat{B}^{\times}$ with $k(b,\omega_{\varphi})\neq0$ such that

Assume that $E(F)$ has rank $1$, choose a generator $P_0$ of $E(F)$ modulo torsion, and for each admissible totally positive $t\in O_F$ let $P_t\in E(F)$ be the trace of the Darmon…

Let $\mathcal{O}$ be the quadratic order used to define the Darmon points, let $H$ be its narrow ring class field over the real quadratic field $K$, and let $J^{(\ell)}_\pm$ be the…

Let $\ell$ be a prime, let $K_\ell$ be the unramified quadratic extension of $\mathbb{Q}_\ell$, and let $T_\epsilon/L_\epsilon$ be the rigid analytic quotient associated with the s…

Let $A_{/\mathbb{Q}}$ be an elliptic curve of conductor $N_A$, let $K$ be a real quadratic field of discriminant $delta_K$ with $(N_A,delta_K)=1$, and let $ell$ be a prime inert in…