10 problems

Eventual p-adic periodicity conjecture. The numbers $\mathcal{E}_{mpn}^{(mp,j)}$ are $p$-adic integers, and for every positive integer $r$ there is a positive integer $n_0$ such th…

Beukers's conjecture. If the representation contains $\alpha$ occurrences of the digit $5$, then

Let $\{W_n\}_{n\geq 0}$ be the Lehmer numbers, defined by … where $\omega=-1/2+\sqrt{-3}/2$. Komatsu–Liu's conjecture. For any nonnegative integers $n,m$ and positive integer $k$,…

Let $f(\tau)=\sum_{n\geq 1}a(n)q^n$ be a holomorphic modular cusp form on a noncongruence subgroup of $\operatorname{SL}_2(\mathbb Z)$. Atkin–Swinnerton-Dyer conjecture. The Fourie…

Kalinin's conjecture. If $p\equiv1\pmod 4$, then

Unit-root supercongruence conjecture. If $a_p(f)\neq 0$, then

Product supercongruence conjecture. For primes $p\geq 5$,

Zudilin's conjecture. Theorem $\mathrm{WIN2a}$ should hold modulo $p^3$.

The 5-adic congruence conjecture. For $p=5$, $m\geq1$, and odd $n\geq1$, one has $c_{5^m}(f)\neq0$ and

Let … ; suppose the cusp at … -rational with cusp width … , let $S_{}$ have dimension … , the space $S_{}$ has a $p$-adically integral basis $f_j(z)=$, depending on $p$, whose Four…