58 problems

Let … be the unique normalized cusp form of weight $12$ for $SL(2,\mathbb{Z})$, with Fourier coefficients $\tau(n)$. Lehmer's conjecture. … The conjecture asserts that the Ramanuja…

Nonvanishing conjecture. The first Fourier coefficient satisfies

Let $f\in S_k(N)$ be a non-CM newform of weight $k\ge 4$, and let $a(p)$ denote its normalized Fourier coefficient at a prime $p$. Atkin–Serre conjecture. For every $\varepsilon>0$…

For even weight $w$ and depth $s$ with $0\leq s\leq w/2$, $s\neq w/2-1$, let $X_{w,s}=\sum_{n\geq 0}a_nq^n$ be a normalized extremal quasimodular form of weight $w$ and depth $s$,…

Let $F\in S_k(\operatorname{Sp}_4(\mathbb{Z}))$ be orthogonal to forms of classical Saito–Kurokawa type, and let $a(T)$ denote its Fourier coefficients for positive-definite half-i…

Divisor-bound conjecture. The coefficients satisfy

Let $\mathcal O$ be a fixed closed orbit, and let $a_{n,\lambda}$ be the coefficients in the expansion associated with $\mathcal O$ and the representation parameter $\lambda$. Clos…

Let $Y$ be a compact hyperbolic surface, let $\phi_i$ be a Laplace eigenfunction with eigenvalue $\mu_i$, and let $a_n^{\sigma/\ell}(\phi_i)$ denote the Fourier coefficient associa…

Let $F = \sum_{n \ge n_0} a_n q^n$ be a positive quasimodular form, and let $\{n: a_n > 0\}$ be the set of indices with positive Fourier coefficient. Its natural density is the lim…

Unbounded denominators conjecture. The Fourier coefficients of $f$ have unbounded denominators. Equivalently, for every positive integer $n$, the Fourier coefficients of $nf$ are n…

For weights $k$ divisible by $4$, let $A(k)$ denote the least positive integer such that every normalized modular form of weight $k$ with nonnegative Fourier coefficients has a pos…

Let $f$ be the cuspidal Hecke eigenform and let $F_f$ be its Gan–Gurevich lift. For a Fourier coefficient indexed by $w$, write $C^{\mu_{\mathbf f}}_w(F_f)$ for the corresponding c…

Kohnen–Skoruppa conjecture. It is conjectured that

Partial-sum conjecture. It is conjectured that

Let $A$ and $B$ be positive integers such that … Write $\mathcal{J}^*_{10}(m)$ for the $m$-th Fourier coefficient of $j^*_{10}(\tau)$. Asymptotic parity conjecture. There are such…

Murty's bounded-repetition conjecture. For fixed $f$ and $a$, $\pi_{f,a}(X)$ is bounded as $X\to\infty$. The paper reports computations consistent with bounded repetition over the…

Let $D_1,D_2$ range over fundamental discriminants with $|D1D2|to obreakter x$, and let $eta(D1,D2)$ be the smallest prime $p$ such that the sign of the corresponding Fourier coeff…

Let $F$ be a Siegel cusp form of degree $2$ and level $N=1$ lying in the Maass subspace, and write $a(T)$ for its Fourier coefficients. Das–Kohnen conjecture. For every…

Let $F$ be a Siegel cusp form of degree $n$, and let $\beta$ be the exponent occurring in the Fourier-coefficient bound denoted by (fc0) in the source. Resnikoff--Saldaña's conject…

Ramanujan–Petersson conjecture. The exponent $\vartheta=0$ is admissible. Unconditionally, the source states that the currently known bound is $\vartheta\leq 5/14$, so the conjectu…

Let $\{A(m,1)\}_{m\geq 1}$ be the sequence of Hecke eigenvalues of a self-dual Hecke–Maass cusp form, and let $\#\mathcal X$ denote the cardinality of a set $\mathcal X$. Positive-…

Let $k>4$ with $k\equiv 2$ modulo $4$. For every reduced matrix $T\in\halfint^+_2$ with bottom-right entry $m$, let $S_m(n,T)$ be the set of integers $r$ such that…

Let $f_1$ and $f_2$ be the eigenforms under consideration, with Fourier coefficients $a_1(p)$ and $a_2(p)$ at primes $p$. Define … and let … where $\delta(\ell)$ is the density of…

Let $F$ be a Siegel Hecke eigenform of degree $2$, and let $A(F,I_2)$ denote its Fourier coefficient indexed by the $2\times2$ identity matrix. Let $L(s,F,\operatorname{spin})$ be…

Orbit Conjecture. The set ${\mathcal O}(\Theta_{2l}^{(r)})$ is the singleton $\{{\mathcal O}_c(\Theta_{2l}^{(r)})\}$. The vanishing assertion is known, while the required nonvanish…