188 problems

Let $E$ be an Anderson $A$-module over a field $R$, and let $b(E)$ be an exponent such that the residue-in-$\tau$ pairing factors through $R^{1/q^{b(E)}}$. Let $w(E)$ range over th…

Let $K=\mathbb{C}(B)$ be the function field of a smooth projective curve $B$ over $\mathbb{C}$. Let $X$ be a projective variety over $K$ that does not contain any possibly singular…

For an integer $n$ written as $n=\delta+h(q-1)$ by Euclidean division by $q-1$, define … and let $P_n(x)$ be … Let $\zeta_C^*(-n)$ be the first nonzero coefficient in the expansion…

Let $Z(x,n)$ be written uniquely as … in $A[\![x-1]\!]$, with $h_n\geq 0$ and $\zeta_C^*(n)\in A$. Let $\operatorname{Cl}(\underline{A}(n))$ denote the class module of the Carlitz…

Banks–Martin conjecture analogue. For each $q\geq 5$, the inequalities

Failure of the local-global principle. The local-global principle for isotropy of quadratic forms fails to hold in dimension $2^n$ over $K$.

Let $K$ be an algebraically closed field and let $F$ be an extension field of $K$. Let $S$ be a finite-dimensional $K$-subspace of $F$ such that $K\subset S$. Define the combinator…

Let $\overline{k}$ be a fixed algebraic closure of $k$, let $k^{\rm sep}$ be its separable closure, and let $G=\operatorname{Gal}(k^{\rm sep}/k)$. Let $\rho:G\to\mathbf C_\infty^*$…

Let $L(x,y)=\prod_i(1-\beta_i^{(y)}/x)$, where the nonzero $\beta_i^{(y)}$ are the zeroes of $L(x,y)$, and let $\lambda_i^{(y)}:=1/\beta_i^{(y)}$ be the reciprocal zeroes. The char…

Let $L(s)$, with $s=(x,y)\in S_\infty$, be a characteristic-$p$ $L$-series arising from arithmetic, and let $\mathbf K_L$ be the finite extension of $\mathbf K_{\mathbf V}$ generat…

Let $I$ be the augmentation ideal of $\mathbb{Z}[\Gamma]$, let $\epsilon$ be the element supplied by the Rubin–Stark conjecture, and let $\Phi\in\iota(\Lambda^n U^*)$. Let…

Let $\mathrm{Gamma}$ be the Galois group in the setup, let $U$ be the relevant group of $S$-units, and let $\mathcal{R}_{\eta}$ be the regulator map associated with…

Let $k$ be a global function field, let $w$ be any place of $k$, and let $N_w$ denote the non-classical set for the interpolation of $L(\chi,s)$ at $w$; when $w=\infty$, this inter…

Let $K$ be a function field of transcendence degree $1$ over $\mathbb{F}_p$, and let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module. Let $n\ge 1$ and let $X$ be an irreducible…

Algebraicity criterion. The generalized double Laurent series is algebraic over $\mathbb{F}_q(t,u)$ if and only if: (a) for some positive integers $a$ and $b$,…

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $E/\mathbb{F}_q(T)$ be an elliptic curve, and let $P,Q\in E(\mathbb{F}_q(T))$ be nontorsion points. Positive common-…

Let $K$ be a characteristic-zero function field, let $E_1/K$ and $E_2/K$ be elliptic curves, and let $P_1\in E_1(K)$ and $P_2\in E_2(K)$ be $K$-independent, meaning that no nonzero…

Let $f:X\to C$ be a fibration over a function field, let $r_K:X\to R(X_K)$ be its rational quotient, and call $X_K$ dominated when it admits the dominant product-type rational map…

Let $f:X\to C$ be a fibration over a function field with $K$-core factorization $f=g_f\circ c_f$, and suppose the orbifold fibration … is not split. Conjecture IV{FF}. There should…

Let $f:X\to C$ be a fibration over a function field, with $K$-core factorization $f=g_f\circ c_f$ and base orbifold $(C(f)/\Delta(c(f)))$. Conjecture III{FF}. The $K'$-rational poi…

Let $K={\bf F}_q(T)$, let $G/K$ be an isotropic, semisimple, simply-connected algebraic group, and let $Z$ be its center. Let ${\bf A}^{\mathrm{fin}}_K$ denote the finite adeles of…

Let $q=19^3$, let … and let $[5]=T^{q^5}-T$ and $M_5$ be Thakur's polynomial. The polynomial $P\in\mathbf{F}_{19^3}[T]$ is the degree-five $c$-Wieferich prime exhibited in the pape…

Let $K=\mathbb C(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety over $K$ containing no rational curve. Let $h_L$ be a Weil height attached to an a…

Let $K=k(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety of general type over $K$. A sequence of $K$-rational points in $X$ is generic if every inf…

Let $K/k$ be a finitely generated extension of fields of characteristic zero, with $k$ algebraically closed in $K$. Define the algebraic special set $\operatorname{Sp}(X)$ as the Z…