9 problems

Algebraicity conjecture. If all Fourier coefficients of the holomorphic part of $\tilde f$ are algebraic, then $\xi(\tilde f)$ is either a unary theta series with…

Let $\varphi$ be an odd ray class group character of conductor $\mathfrak{m}\cdot\infty_1$, let $f_\varphi$ be the associated holomorphic weight-one cusp form, and define…

Transcendence conjecture. The holomorphic coefficients of $f$ are conjectured to be transcendental almost always.

Ono–Bruinier conjecture. All but a set of density $0$ of the non-vanishing coefficients of the holomorphic part of $f$ should be transcendental.

Let $m>0$ and let $f$ be a harmonic weak Maass form of weight $\frac{1}{2}-m$ for the representation $\rho_L$. Assume that $\xi_{\frac{1}{2}-m}f$ is holomorphic also at $\infty$, a…

Goldfeld's conjecture. A positive proportion of discriminants $\Delta$ satisfying $0<|\Delta|\leq X$ have

Let $\Delta$ and $r$ be as in the preceding notation, let $E_\Delta$ be the corresponding elliptic curve with minimal Weierstrass model $W_\Delta$, and let…

Let $E$ be the elliptic curve over $\mathbb{Q}$ corresponding to the newform $G$, let $\mathcal{W}$ be a minimal Weierstrass model of $E$ over $\mathbb{Z}$, and let the differentia…

Let … be a weight $1/2$ harmonic weak Maass form on $\Gamma_0(N)$ whose principal parts at cusps are defined over a number field. Let $\xi_{1/2}(f)$ be its image under the usual di…