25 problems

Let $K$ be the base field, let $\Omega$ be the domain of the Drinfeld modular forms, and let $t(z)$ be the usual parameter at infinity for $z\in\Omega$. The functions $E$, $g$, and…

Root-exponent bound conjecture. If $g_{\underline{r}}$ has a $k$-th root as a modular function on $X_0(\mathfrak{n})$, then

Characterization conjecture. This criterion characterizes all Drinfeld modular units on $X_0(\mathfrak{n})$.

Gekeler–Rahm residue homomorphism conjecture. The residue homomorphism is an isomorphism for $r=3$.

Nonvanishing conjecture. This element is non-zero in $\mathbb{T}(\mathfrak{n},\mathbb{F}_p)$. Consequently, the pairing $\varphi_{\mathrm{DMF}}$ is not perfect.

Let $A=\mathbb{F}_q[T]$, let $mathfrak p\trianglelefteq A$ be a maximal ideal of degree $d$, and let $n\geq 1$. For integers $k,l$, write $\mathbf{T}_{mathfrak p}$ for the correspo…

Let $A=\mathbb{F}_q[T]$ and let $\mathrm{S}_{k,l}$ denote the space of Drinfeld cusp forms of weight $k$ and type $l$. The operator $\mathbf{T}_T$ is the Hecke operator associated…

Non-square congruence subgroup conjecture. Both $\Gamma_1(\alpha T+\beta)$ and $\Gamma_0(\alpha T+\beta)$ may be non-square congruence subgroups for any choice of $\alpha\neq 0$ an…

Let $A$ be the coefficient ring and let $T$ be the prime defining the level. For weights $k,l$, write $S_{k,l}(\mathrm{GL}_2(A))$ for the space of Drinfeld cusp forms for…

Let $M_{k,l}(\Gamma_0(\mathfrak{n}))$ denote the space of Drinfeld modular forms of weight $k$ and type $l$ for the congruence subgroup $\Gamma_0(\mathfrak{n})$. An integral basis…

Let $A=\mathbb{F}_3[t]$, let $n\geq 2$, and set $k_n=1+3^n$. Let $\Sigma_n\subset(\mathbb{F}_3^\times)^n$ be the set of sign tuples with exactly $i$ entries equal to $-1$ for some…

Let $k$ be the weight of a Drinfeld modular form of level $\Gamma_0(T)$, and let the slope of a $\mathrm{U}$-eigenvalue mean its valuation. A form is new if it belongs to the suita…

Modular analogue conjecture. For all $m\geq0$, if $|\Sigma|=s=m(q-1)+1$ and $q>m$, then

Multiple Eisenstein correspondence conjecture. The following properties hold: (1)…

The correspondence conjecture. The correspondence $\zeta_A(\mathcal{C})\leftrightarrow\varphi_A(\mathcal{C})$ induces an isomorphism of $\mathbb{F}_p$-algebras

Let $t$ be the prime-level polynomial denoted by $P_1=t$. Let $\mathbf{T}_t$ be the operator on the relevant space of Drinfeld cusp forms, and let $S^1_{k,m}(\Gamma_0(t))$ denote t…

Refined slope conjecture. At weight $k'$, the old slopes at weight $k$ occur with exactly the same multiplicity: for every old slope $\alpha$ in weight $k$,

Slope multiplicity conjecture. Under these hypotheses,

The oldform and Hecke conjectures. The following assertions should hold:

Joshi–Petrov conjecture. Given $q$, there exist $k\gg 0$ and $m$, with $0\leq m<q$ and $k\equiv 2m\pmod{q-1}$, such that

Let $\mathbb{T}_{>0}$ be the algebra of power series with positive convergence radius, and let $g$, $h$, and $\boldsymbol{s}_2$ be the almost $A$-quasi-modular forms used in the so…

Let $\mathbb{T}_{>0}$ be the $C$-algebra of power series with positive convergence radius, let $\widetilde{\mathcal{M}}$ be the algebra of almost $A$-quasi-modular forms, and let…

Let $q=p^e$ be a prime power, let $A=\mathbb{F}_q[\theta]$, and let $f$ be a non-zero Drinfeld quasi-modular form of weight $w$ and depth $l\geq 1$. Write $\nu_{\infty}(f)$ for its…

Let $K$ be the coefficient field, let $\Omega$ be the Drinfeld upper half-plane, and let $u$, $E$, $g$, and $h$ be respectively the parameter at infinity, the False Eisenstein seri…

Let $f$ be a differentially extremal element of the algebra $\widetilde{M}$ of Drinfeld quasi-modular forms. A form is defined over a field $L\subset C$ if $f\in L[E,g,h]$, where…