16 problems

Let $p>2$ be a prime, let $E/{\mathbb Q}$ be an elliptic curve, and let $\Sigma_p$ denote the set of Dirichlet characters of order $p$. For $S\subset\Sigma_p$, say that $S$ has den…

Let $(a,b,m,n)\in\mathcal{M}$ be a Master-Hit, and let $f_1$ and $\mathcal{P}(a,b,m,n)$ denote the associated integer and set of primes. For a prime $\ell$, write $v_\ell(f_1)$ for…

Let $X\to B$ be as in Theorem MMP, with $X$ a Calabi–Yau variety. Suppose that $B$ is birational to a Mori fiber space $B'\to U$ with fiber $F$, and let…

Non-rationality conjecture. The specializations are not rational in $t$ with coefficients in $K_0(\operatorname{Stck}_{\mathbb{C}})[\mathbb{L}^{-1}][v]$ and…

Let $p\geq7$ and let $(p,p-2)$ be a twin-prime pair. Define the twin-prime elliptic curve by … Write $r(E_p)$ for its Mordell-Weil rank. Corrected twin-prime rank conjecture. … The…

The Diophantine exponent conjecture.

Let $\mathcal{A}$ be a simple abelian variety of dimension $g$ over a number field $K$ embedded in $\mathbb{R}$. Let $\mathcal{A}(\mathbb{R})^0$ be the identity component, let…

Rank-zero fibre conjecture. There are infinitely many $b\in\mathbb{P}^1(\mathbb{Q})$ such that

Non-isotrivial elliptic-surface rank conjecture. If $\pi$ is non-isotrivial, then both $\mathscr{J}(X,\pi)$ and $\mathscr{N}(X,\pi)$ are infinite.

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $\mathbf L\subset\mathbb{Q}^{\operatorname{ab}}$ be a real abelian field containing only finitely many extensions of…

Cowan's conjecture. For every pair of positive integers $m,n$,

Let $E$ be the elliptic curve under consideration, and let $F/\mathbb{Q}$ be an infinite real abelian extension containing only finitely many subfields of degree $2$, $3$, or $5$ o…

Let $\mathbb{Q}_n$ be a finite cyclotomic layer, let $\chi$ be a character of $\operatorname{Gal}(\mathbb{Q}_n/\mathbb{Q})$, and let $I_\chi$ be the kernel of the induced map from…

Larsen's conjecture. Then

Let f\in S_k(\Gamma_0(pN^+))^{p\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-new}} be a $p$-new eigenform, let $L_{f,\mathfrak{p}}$ be the completion of its…

Mazur's conjecture. One of the following two conditions holds: