47 problems

Let $E_t^d$ be the CM elliptic curve and let $w_d$ denote the order of the relevant unit group. For a prime $p$, write $\left(\frac{a_p(E_t^d)}{p}\right)_m$ for the $m$th power res…

CM residue-disk placement conjecture. The $j$-invariants of all such curves $E$ lie in $2(p+1)/i$ residue disks inside the corresponding Atkin–Lehner circle, with $(p+1)/i$ disks f…

CM placement conjecture at 13. If

CM placement conjecture at 7. If

David's conjecture. For every $\varepsilon>0$, there exists a constant $c(A/K,\mathcal{L})>0$ such that

Let $X$ be the once-punctured elliptic curve under consideration, let $K$ be its complex multiplication field, let $I_K$ be the specified index set, and let $\boldsymbol{m}$ range…

A variety is said to be of CM type when it has the complex multiplication property described in the surrounding theory. Piatetski-Shapiro–Shafarevich conjecture. Varieties of CM ty…

Let $G_{k,D}^{(r)}$ be the CM higher-order-pole forms, let $k\in\{4,6,8,10,14\}$, $1\leq r\leq k-1$, and let $D<-4$ be a discriminant with class number $1$. The complementary-pole…

Let $G_{k,D}^{(r)}$ denote the integral linear combination of $E_k/(j-j(\alpha_D))^i$ for $1\leq i\leq r$ having a pole of order $r$ at the CM point of discriminant $D$. Let…

Let $w$ be a prime of $\mathbb{Q}^{\mathrm{al}}$ and let $A$ be an abelian variety over $\mathbb{Q}^{\mathrm{al}}$ with good reduction at $w$. Let $\gamma$ be a Hodge class on $A$.…

Let $f(q)=\sum_{n=1}^\infty a_nq^n\in S_2(\Gamma_0(N))^{\operatorname{new}}$ be a normalized newform of CM type mod $\ell$ by an imaginary quadratic field $K$. A newform has comple…

Koblitz's conjecture. There exists a constant $C>0$ such that

Let $\mathcal{M}_g$ denote the moduli space of smooth curves of genus $g$. A curve in $\mathcal{M}_g$ has a CM Jacobian when its Jacobian is a CM abelian variety. Coleman–Oort conj…

Let $d_1$ and $d_2$ be discriminants such that one of them is fundamental when $k$ is even. For any CM points $z_1$ and $z_2$ of discriminants $d_1$ and $d_2$, let $H$ be the compo…

Let $C$ be the smooth plane curve … Let $J$ be its Jacobian, let $K=\mathbb{Q}(\zeta_9)^+(\zeta_5)$, and let $\mathcal{O}_K$ denote the ring of integers of $K$. Somoza's endomorphi…

Let $\ell$ be a prime, let $K$ be a quadratic number field, and let $E/K$ be an elliptic curve. An elliptic curve is heavenly at $\ell$ when its $\ell$-power torsion field satisfie…

Let $\tau\in\mathbb{H}$. The numbers … are considered, where $E_2$, $E_4$, and $E_6$ denote the usual Eisenstein series. Nesterenko's conjecture. If at most three of these five num…

Residue conjecture. The residues of $\tau^m f(\tau)$ satisfy

Let $E/\mathbb{Q}$ be a CM elliptic curve, let $r\neq 0$ be a fixed integer, let $C_{E,r}$ be the Lang–Trotter constant, and let $\overline{\omega}_{E,r}$ be the explicit constant…

Let $m$ be a positive odd integer, and let $J_m$ denote the Jacobian associated with the curve $C_m\colon y^2=x^m-1$. Write $\phi(d)$ for Euler's totient function. The simple-facto…

Gross's conjectural formula. The order of the Tate–Shafarevich group of the Gross curve over $H$ is

Class-polynomial conjecture. The polynomial $H_{D,N,m}(X)$ belongs to $\mathbb{Z}[X]$ and is irreducible. This is proposed as an analogue of the preceding classical class-polynomia…

Let $k\geq 2$, let $f\in M^!_{2-2k}$ be a weakly holomorphic modular form with rational Fourier coefficients, and let $G_{k,f}$ be the higher Green's function associated to $f$. Wr…

Let $q$ be a prime with $q \equiv 3 \ \mathrm{mod}\ 8$, let $K=\mathbb{Q}(\sqrt{-q})$, let $H$ be the Hilbert class field of $K$, and let $A$ be the Gross elliptic curve over $H$.…

Let $d_1$ and $d_2$ be the discriminants under consideration, with $d1 e d2$ and $d1 e 0$, $d2 e 0$. Assume $d1 \equiv d2 \equiv 5\pmod 8$. For each $t=\frac{m+\sqrt D}{2}inmathcal OF$,…