10 problems

Modularity conjecture. The function

Assume the height pairing on Heegner cycles is positive definite. Let … Here $Z_\kappa(f)$ is the Heegner cycle associated with the harmonic weak Maass form $f$ attached to the nor…

Arithmetic modularity conjecture. The function $\widehat{\phi}_\kappa$ is a cusp form. For $\kappa=1$, the source states that arithmetic modularity on the modular curve $X_0(N)$ ha…

Let $\mathrm{Pin}^{\mathrm{alg}}$ be the subgroup of algebraic principal cycles and let $\mathrm{Pin}$ be the subgroup of principal cycles in the group of Heegner cycles. Principal…

Let $G\in S_{2\kappa}^{\mathrm{new}}(N)$ be a normalized newform, let $F_G$ be the totally real number field generated by the eigenvalues of $G$, and let…

Let $G\in S_{2\kappa}^{new}(N)$ be a normalized newform, let $F_G$ be the totally real field generated by its eigenvalues, and let…

Heegner cycle Main Conjecture. The following hold:

Let $p$ be inert in $K_D=\mathbb{Q}(\sqrt{D})$, let $2k>2$ be an even weight, and let $\mathfrak Z^\mathrm{How}_{D,r,2k}$ and $\mathfrak Z_{D,r,2k}^\mathrm{Heeg}$ denote respective…

Let $(f,K,\mathfrak p)$ be an admissible triple. Let $K_\infty$ be the anticyclotomic $\mathds Z_p$-extension of $K$, let $\Lambda={\mathcal O}_\mathfrak p[[\Gamma_\infty]]$ be its…

Let $f$ be a modular form of even weight $r+2$, let $\psi$ be an unramified algebraic Hecke character of an imaginary quadratic field $K$ of infinity type $(r,0)$, and let $W$ be t…