12 problems

Let $\mathbb{K}$ be a finite field, let $d\geq 2$ be an integer, and let … Here $\operatorname{PR}_{\mathbb{K}}(f)$ is the partition rank and $\operatorname{AR}_{\mathbb{K}}(f)$ is…

Selmer criterion for plectic Heegner classes. If $\chi$ is primitive, then

Weak Goldfeld conjecture. For each $r\in\{0,1\}$,

For each prime $p$, let $\operatorname{perdim}(p)$ be the number of weight-$2$ newforms $f$ for $\Gamma_0(p)$ such that $w(f)=-f$ and $L(f,1)=0$. Positive-even-rank dimension conje…

For each prime $p$, let $\operatorname{strdim}(p)$ denote the strangeness dimension defined in the surrounding discussion. Strangeness-dimension conjecture. The following two forms…

Lovett–Adiprasito–Kazhdan–Ziegler conjecture. There exists a constant $C_d$ such that for any finite field and multilinear form $f$ we have

Constant-factor equivalence conjecture. For every $k ge 2$ there is a constant $C=C(k)$ such that, for every finite field $F$ and every $k$-tensor $T$ over $F$,

Let $d\geq 2$, let $\mathbb F_q$ be a finite field, and let $P$ be a multilinear polynomial of degree $d$. Denote by $r_{\mathbb F_q}(P)$ its Schmidt rank and by…

Consider weight $2$ newforms, ordered by level, and call a newform’s degree the degree of its rationality field over $\mathbb Q$. Degree-one analytic-rank conjecture. Among all wei…

Let $K$ be the relevant number field, let $K_p$ denote its $p$-adic localization, let $\mathbb{T}$ be the associated big representation, and let $r_{\mathrm{an}}(\mathscr{M})$ be t…

Let $K$ be a number field, let $E$ be an elliptic curve over $K$, and let $C(K,X)$ denote the set of quadratic characters of $K$ of norm less than $X$. Write $E^\chi$ for the quadr…

Darmon's conjecture. The point $P_\varphi$ belongs to $E_0(H_{\mathfrak c})$, the point $P_K$ belongs to $E_0(K)$, and $P_K$ is non-torsion if and only if