547 problems

Determinant conjecture.

Let $n\geq 0$, $i\geq 3$, and $k\geq 0$. The overcubic partition tuple function $\overline{b}_m(r)$ is considered at the indicated arithmetic progressions. Congruence conjecture. F…

Let $p_r(n)$ denote the coefficients in the partition-generating series used in the paper. For a positive integer $\lambda$, let $\mu_{\lambda}$ and $\delta_{\lambda}$ satisfy … Hi…

Let $d_k(n)$ denote the $k$-elongated plane partition function, and let $c,n\geq 0$. For a list of integers in an argument, interpret the corresponding congruence as holding for ea…

Further generalized cubic partition congruence conjecture.

For every smooth projective surface $S$ with numerically trivial canonical class, and for all tautological insertions (in particular, line bundles $L_1,\ldots,L_N$ and indices…

Given a discrete subgroup $\Gamma\leq \operatorname{PSL}_2(\mathbb{Z})$ in Jones's cusp-form and von Neumann-algebra construction, find an explicit formula for a tracelike vector…

For every modular knot $K$, construct explicitly a weight-$2$ modular integral $F_K$ for $\mathrm{SL}_2(\mathbb{Z})$ such that, for every modular knot $L$, the homogenized cycle in…

For each $i\in\{1,2,3,4,5\}$, prove the corresponding modulo-$9$ Kanade–Russell Rogers–Ramanujan-type sum–product identity, namely…

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathbf{T}_r$ be its Cartan matrix. For $\mathbf{x}_r=(x_1,\ldots,x_r)$, define the tadpole Nahm sum … The sum…

Let $\bold Q$ denote the field of rational numbers, and call an elliptic curve over $\bold Q$ semistable if it is semistable at every prime, meaning that at each prime it has eithe…

Let $\bold Q$ denote the field of rational numbers, and call an elliptic curve over $\bold Q$ modular if it admits a holomorphic map from some modular curve $X_0(N)$ onto it. Taniy…

Mazur's main conjecture for modular forms.

Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, let $\Lambda$ be its Iwasawa algebra, and assume that $\rho_f$ has large image. For a height-one…

Nonvanishing conjecture. The first Fourier coefficient satisfies

Let $f$ be a holomorphic cusp form with associated quadratic-twist family $\mathcal{F}_f^{+}(X)$, and let $R_f(\alpha,\gamma)$ denote the average of the ratio of shifted $L$-functi…

Maeda's conjecture. The Galois conjugates satisfy

Kolyvagin's conjecture. The set of Kolyvagin classes is nontrivial:

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term. Odd-prime valuation conjecture. If $p>2$ is prime and $a$ is a positive integer, then … This is an empiric…

Generalized Atkin–Serre conjecture. Under these hypotheses, the stated lower bound holds for every nonzero Fourier coefficient $a_f(m)$. This extends the Atkin–Serre conjecture fro…

Let $t_o(n)$ be defined by … where $q$ is a complex number with $|q|<1$ and $(a;q)_n$ denotes the standard $q$-Pochhammer symbol. Andrews–El Bachraoui's conjecture. If $8n+9$ has a…

Let $F$) be a totally real field and let … be a continuous, irreducible and totally odd representation. Let $D$ be a quaternion algebra over $F$ that splits at exactly one infinite…

Suppose that $a_p(f)=0$ and $\psi=\mathbf{1}$. Let $\mathrm{Sel}^{\pm}(\mathbb{Q}_\infty,A_f(1))^\vee$ be the Pontryagin duals of the signed Selmer groups, let…

Modularity conjecture. The function

Let $(X,H)$ be a smooth polarized surface with $H_1(X,\mathbb{Z})=0$ and $h^{2,0}(X)>0$. Let $r$ be prime and let $c_1\in H^2(X,\mathbb{Z})$ be algebraic. Let…