50 problems

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…

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…

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

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…

Goldfeld's moderate-growth conjecture. The modular symbol has moderate growth uniformly when $f$ is allowed to vary among the normalized newforms; that is, there are constants…

Let $f$ be a holomorphic cusp form of weight $2$ for a fixed cofinite discrete subgroup $\Gamma$ of $\operatorname{SL}_2(\mathbb R)$, and let $\Gamma_{\!\infty}$ be the stabilizer…

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 an elliptic curve, let $p$ be a prime, and let $\widetilde{\delta}^{(1)}_n$ denote a mod-$p$ Kurihara number. Kurihara's mod-$p$ Selmer-structure conjecture.…

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, let $E$ be a real quadratic field, and let $H(\Gamma,E)$ denote the image of the connecting homomorphism…

Let $p$ be a prime, and consider normalised modular symbols attached to weight $2$ cusp forms, reduced modulo $p$. Mazur–Rubin's equidistribution conjecture. The normalised modular…

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 $E$ 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…

Mazur–Tate conjecture. For any $\psi\in\operatorname{Symb}_{\Gamma_0}(R)$ such that $T_q\psi=a_q\cdot\psi$ for all primes $q$, one has in $R$

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 $\tilde{\mathcal{M}}_{\Gamma}$ be the relevant modular-symbol module and let … be the anti-symmetric pairing on its $\mathbf{Z}$-dual $\tilde{\mathcal{M}}_{\Gamma}^*$ construct…

Let $f$) be a cusp form for $\Gamma_0(N)$, let $M>0$, and for $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ define $|\\!|\gamma|\\!|_z:=|cz+d|^2$. The modular symbol is denoted by…

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…