23 problems

Let $E$ be a fixed elliptic curve over $\mathbf{Q}$, and let $E^d$ range over its quadratic twists. The analytic rank is the order of vanishing of $L(E_d,s)$ at the center of the c…

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

Let $K$ be a number field and let $\Gamma$ be a congruence group such that the moduli curve $\mathcal{X}_{\Gamma}$ is isomorphic to $\mathbb{P}^1$ over an open subset of…

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

Analytic rank–partition rank conjecture. For every $d\geq 2$, there is a constant $C_d$ such that

For $d\neq 0,\pm3$, let $C_d$ be the genus-$2$ curve in the paper's displayed family, and let $g_d$ denote its analytic rank. Let $S(X)$ be the set of positive square-free integers…

Let $C$ be a hyperelliptic curve defined by $y^2=f(x)$. For $k\in\mathbb{Q}/(\mathbb{Q}^\times)^2$, let $C^{(k)}$ be the hyperelliptic twist defined by … The analytic rank of…

For a multilinear map, let the partition rank measure its decomposition complexity and let the analytic rank measure its randomness over a finite field. Gowers–Wolf conjecture. The…

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…

Adiprasito–Kazhdan–Ziegler conjecture. The bound in Theorem $$ should hold with linear bounds; equivalently, the relevant strength and partition-rank quantities over $K$ should be…

Strength–analytic-rank conjecture. Strength is bounded by a constant multiple of analytic rank:

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

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $D$ range over quadratic discriminants, and let $C_E$ and $b_E$ be constants depending on $E$. Conjecture on the frequency of hi…

Let $f$ be a newform, let $\mathds Q_f$ be its Hecke field, and for every embedding $\sigma:\mathds Q_f\hookrightarrow\mathds C$ let $f^\sigma$ be the conjugate newform. Galois-inv…

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…

Let $E$ be a fixed elliptic curve over $\mathbb{Q}$ and let $l\geq 7$ be a fixed prime. Let $\chi$ range over primitive Dirichlet characters of order $l$, and let…

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…

Analytic-rank growth conjecture. The analytic rank is unbounded as $E$ varies over elliptic curves over $\mathbb Q$, equivalently, the paper's stated goal concerns the conjecture t…