36 problems

Let $\xi$ and $\xi'$ be the cycles associated with Algorithm, and suppose $n\leq4$ and $\|\xi\|>1$. For each modular symbol ${\mathbf v}$ in step A of the algorithm, choose…

Let ${\mathbf v}=[v_1,\dots,v_n]$ be a modular symbol with $\|{\mathbf v}\|>1$, and define … Let $E$ be the standard basis of ${\mathbb Z}^n$. The quadratic form $b({\mathbf v})$ i…

Let ${\mathbf v}=[v_1,\dots,v_n]$ be a modular symbol with $\|{\mathbf v}\|>1$. Define … For a top-dimensional cone $\sigma$ in the Voronoi decomposition containing…

Let $M^*_{(m)}$ be the modular complex, let $\Gamma_1(m;N)$ be the indicated congruence subgroup, let ${\rm V}_m$ be the representation appearing in the source, and let…

Let $R_j^+(n)$ denote the scalar quantity associated with the embedded spectral contribution, and let $r(X)$ be the corresponding error term in the twisted Ramanujan-sum asymptotic…

For $m,n\neq0$, define … and let $r(S,S^*)(x)$ be the Pearson correlation of the sequences $S(m,n;c)$ and $S^*(m,n;c)$ over the dyadic window $W_x=\{N\mid c:x<c\leq2x\}$. Uncorrela…

Selberg–Linnik bound. For every $m,n\neq0$ and every $\varepsilon>0$,

Mazur--Tate's refined exceptional zero conjecture. One has

Let $E/\mathbb{Q}$ be the elliptic curve in the setup, let $p$ be an odd prime satisfying the irreducibility and Manin-constant hypotheses, and let…

Sharifi's conjecture. The map $\Pi_N$ factors through a map

Let $p\nmid N\phi(N)$, let $\theta$ be a primitive even Dirichlet character of conductor $Np$, and let $\varpi_{\theta}$ denote the $\theta$-component of Sharifi's homomorphism fro…

Let $f\in S_2(\Gamma_0(q))$, let $c$ tend to infinity through positive integers satisfying $(c,q)=d$, where $d$ divides $q$, and let $a$ range over $(\mathbb Z/c\mathbb Z)^*$. Let…

Let $f\in S_2(\Gamma_0(q))$, where $q$ is a positive integer. For each positive integer $c$, define … where $\langle r\rangle=\int_{i\infty}^r\operatorname{Re}(f(z)\,dz)$ and…

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$ with conductor $N$, and for every real cyclic extension $F/\mathbb{Q}$ of degree $d\geq3$ and conductor $m$ define the norm…

Modular-symbol distribution conjecture. As $m$ tends to infinity, the distribution of the values $\{[a/m]^+/\sqrt{\log(m)}:a\in(\mathbb{Z}/m\mathbb{Z})^\times\}$ converges to a nor…

Let $\widetilde{\varpi}:H_1(X_1(N),C_{\infty},\mathbf Z_p)_+\to\mathcal K$ be the cyclotomic map, and let $\mathfrak I_{\infty}$ be the Eisenstein ideal acting on the source. Shari…

Let $N$ be the prime level and $p$ a prime with $p\nmid N$. Let $H_+$ be the plus part of the modular-symbol homology group, let $I$ be the Eisenstein ideal acting on $H_+$, let…

Let $G$ be a nontrivial finite abelian group and $n\ge 1$. Let $H_{n,\mathrm{prim}}^{BM}(\mathbb X_n,\mathcal L_{n,G})$ be the common kernel of the nontrivial co-multiplication map…

Let $G$ be a nontrivial finite abelian group and $n\ge 1$. In the Borel–Moore homology $H_n^{BM}(\mathbb X_n,\mathcal L_{n,G})$, let $H_{n,\int}(\mathbb X_n,\mathcal L_{n,G})$ be t…

The Laplacian invertibility conjecture. On each term of the complex, the operator $\bm{\Delta}$ is invertible after tensoring with $\mathbb Q$, except on the first term…

Let $G$ be a finite abelian group, let $\mathcal G_\bullet$ range over flags of subgroups of $G$, and let $\mathcal M_n^-(G)$ denote the antisymmetric symbol module. The flags and…

For each integer $N\ge 2$, consider the symbol $[0,0,1]$ in the module $\mathcal B_3(\mathbb Z/N\mathbb Z)$. The order conjecture. The element has order $1$, equivalently…

Mazur–Rubin–Stein conjecture. For each $x\in[0,1]$, the limits satisfy

Let … be the homomorphism induced by the Manin-symbol map. Let $T_n$ denote the Hecke operators and let $\langle d\rangle$ denote the $d$th diamond operator. Sharifi-inspired away-…

Assume $p\geq5$. Let $L$ be the set of supersingular lambda invariants in $\mathbf{F}_{N^2}$, let $H'$ be the derivative of the associated polynomial $H$, and extend $\log$ to a su…