119 problems

Let $X/k$ be a curve with Jacobian $J$, and write $g(X)=\dim J$. A CM Jacobian is a Jacobian with complex multiplication. Degeneracy can only occur when … Coleman's conjecture. Up…

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

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 $f$ be a modular form and let $V_f$ be its associated two-dimensional $p$-adic Galois representation. Greenberg's local splitting conjecture. If the restriction of $V_f$ to the…

Let $E/\mathbb{Q}$ be an elliptic curve, and let … count amicable pairs of $E$ up to $X$. Assume that there are infinitely many primes $p$ such that $\#E_p(\mathbb{F}_p)$ is prime.…

Let $E$ be an elliptic curve over $\mathbf Q$, let $m\geq 1$, and let $\mathcal C$ be a set of primes contained in the set of primes congruent to $1$ modulo $m$. Write…

Let $K$ be an imaginary quadratic field of discriminant $d_K$ other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, let $\mathfrak{n}$ be a proper nontrivial ideal of its…

Taylor's conjecture.

Let $K={\bf Q}(\sqrt{-D})$ with $D$ prime and $D\equiv 1\pmod{4}$, and let $E$ be an elliptic curve having complex multiplication by ${\cal O}_K$. The Steinitz class $St(E)$ is the…

Meromorphic continuation conjecture. The function $L_m(E,s)$ has an entire continuation, except that when $4\mid m$ and $E$ has CM, it has a pole at

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…

Fix an imaginary quadratic field $K$ of class number one and an elliptic curve $E$ with complex multiplication by $\mathcal O_K$, and let $w$ be the order of $\mathcal O_K^\times$.…

Lauter's denominator-bound conjecture. Every prime dividing the denominator of a coefficient of $H_i(X)$ is bounded by the discriminant of $K$.

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

CM placement conjecture at 5. If

Relative Lehmer conjecture for elliptic curves. There exists a constant $c(E/K)>0$ such that

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

Let $f$ be a rational newform of weight $k\geq 2$ with complex multiplication by $\mathbf{Q}(\sqrt{-d})$, where $d>3$, and let $m$ be relatively prime to $2(k-1)$. CM power-residue…

Let $A$ be an abelian variety of CM type over a number field $k$, endowed with an ample symmetric line bundle $\mathcal{L}$, and let $V$ be a proper $k$-irreducible algebraic subva…

4-division factorization conjecture. Suppose that $D\equiv 7\pmod 8$. Then $f_4$ has three factors of degree $2$ over $\mathbb{Q}(j)$. One of them has discriminant $DF^2$, and the…

Discriminant-square conjecture. If $D\equiv 0\pmod 8$, then $u+64=DZ^2$ for some $Z\in\mathbb{Q}(j(\alpha))$.

Let $Q$ be a finite algebraic extension of ${\bf Q}$ whose embeddings into ${\bf C}$ factor through $Q_0$, let ${\cal M}$ be a homogeneous $Q$-motive over $Q_0$ of weight $k$, and…

Let $X$ be a Calabi–Yau threefold and let $X^\vee$ be its mirror manifold. A Calabi–Yau threefold is of CM type when it has the complex-multiplication property in the sense used fo…