9 problems

Let $f$ be a weight-$2$ newform of level $\Gamma_0(p)$ with integer Fourier coefficients, let $E=E_f$ be the associated elliptic curve with multiplicative reduction at $p$, and let…

Stark–Heegner point conjecture. The points $P_\tau^+$ and $P_\tau^-$ are defined over $H_{\mathcal O_\tau}$ and $H_{\mathcal O_\tau}^+$, respectively, and $P_\tau^-$ belongs to the…

Algebraicity conjecture. If $r_{\mathrm{alg}}(A/E)\geq r$, then there exists $w_{A,S}\in\bigwedge^r(A(E)_{\mathbb Q})$ such that

Let $F$ be a real quadratic field, let $E/F$ be a modular elliptic curve with everywhere good reduction, and let $K/F$ be an almost totally real quadratic extension. With the assoc…

Stark-Heegner Conjecture. The points $P_{\mathfrak a}$ belong to the natural image of $E(H)$ in $E(H\otimes\mathbb{Q}_p)$, and $P_\psi$ is non-trivial if and only if…

Algebraicity and Shimura reciprocity conjecture. If $r_\mathrm{alg}(A/E,\chi)\ge r$, then there exists an element $w_A^\chi\in\wedge^r A(E_\mathfrak{c})^\chi$ such that

Let $\tau$ be an RM point and let $H_\tau$ be its narrow ring class field. The associated Stark–Heegner points in the modular Jacobian $J_0(p)$ are conjectured to be defined over…

Bertolini–Darmon conjecture. Up to multiplication by a non-zero integer, the point $P_{\chi}$ lies in $E(H_{\chi})$, and it is non-torsion if and only if

Let $K$ be a real quadratic field, and let the conductor of an elliptic curve $E_{/\mathbb{Q}}$ factor as $N=pDM$ into relatively prime integers, where $D$ is the square-free produ…