We study the problem of bounding the least prime that does not split completely in a number field. This is a generalization of the classic problem of bounding the least quadratic nonresidue. Here, we present two distinct approaches to this problem. The first is by studying the behavior of the Dedekind zeta function of the number field near $1$, and the second by relating the problem to questions involving multiplicative functions. We derive the best known bounds for this problem for all number fields with degree greater than $2$. We also derive the best known upper bound for the residue of the Dedekind zeta function in the case where the degree is small compared to the discriminant.

We give a geometric interpretation of the inner product on the modified quantum group of ${\widehat{\mathfrak{s}\mathfrak{l}}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.

This paper is a combinatorial and computational study of the moduli space $M_g^{tr}$ of tropical curves of genus $g$, the moduli space $A_g^{tr}$ of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were studied recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the category of stacky fans, of which $M_g^{tr}$ and $A_g^{tr}$ are objects and the Torelli map is a morphism. We compute the poset of cells of $M_g^{tr}$ and of the tropical Schottky locus for genus at most 5. We show that $A_g^{tr}$ is Hausdorff, and we also construct a finite-index cover for the space $A_3^{tr}$ which satisfies a tropical-type balancing condition. Many different combinatorial objects, including regular matroids, positive-semidefinite forms, and metric graphs, play a role.

We prove some results on the structure of certain classes of integral fusion categories and semisimple Hopf algebras under restrictions on the set of their irreducible degrees.

There is a natural ${\mathbb{S}}_n$-action on the moduli space ${\overline{\mathcal{ℳ}}}_{1,n}\left(B{\left(\mathbb{Z}∕m\mathbb{Z}\right)}^2\right)$ of twisted stable maps into the stack $B{\left(\mathbb{Z}∕m\mathbb{Z}\right)}^2$, and so its cohomology may be decomposed into irreducible ${\mathbb{S}}_n$-representations. Working over $Spec\mathbb{Z}\left[1∕m‘\right]$ we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for $\Gamma \left(m‘\right)$. In particular this offers an alternative to Scholl’s construction of the Chow motive associated to such cusp forms. This answers in the affirmative a question of Manin on whether one can replace the Kuga–Sato varieties used by Scholl with some moduli space of pointed stable curves.

Let $k$ be a number field, $f\left(x\right)\in k\left[x\right]$ a polynomial over $k$ with $f\left(0\right)\ne 0$, and ${\mathcal{O}}_{k,S}^{\ast }$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural exceptional set $T\subset {\mathcal{O}}_{k,S}^{\ast }$, the prime ideals of ${\mathcal{O}}_k$ dividing $f\left(u\right)$, $u\in {\mathcal{O}}_{k,S}^{\ast }\setminus T$, mostly have degree one over $\mathbb{Q}$; that is, the corresponding residue fields have degree one over the prime field. We also formulate a conjectural analogue of this result for rational points on an elliptic curve over a number field, and deduce our conjecture from Vojta’s conjecture. We prove this conjectural analogue in certain cases when the elliptic curve has complex multiplication.

On donne des résultats de non-existence pour les points rationnels de la courbe modulaire de Drinfeld affine $Y_1\left(\mathfrak{p}\right)$ avec $\mathfrak{p}$ idéal premier de ${\mathbb{F}}_q\left[T\right]$. Cette courbe classifie les modules de Drinfeld de rang $2$ munis d’un point de torsion d’ordre $\mathfrak{p}$. Le premier énoncé concerne les points définis sur les extensions de ${\mathbb{F}}_q\left(T\right)$ quadratiques pour $\mathfrak{p}$ de degré $3$ et cubiques pour $\mathfrak{p}$ de degré $4$ et $q\ge 7$. Le deuxième, conditionné à une dualité entre algèbre de Hecke et formes modulaires de Drinfeld, concerne les points sur les extensions de degré $\le q$ pour $deg\mathfrak{p}$ suffisamment grand. Comme conséquence, on déduit, sous la même condition, une borne uniforme pour la torsion des modules de Drinfeld de rang $2$ définis sur les extensions de ${\mathbb{F}}_q\left(T\right)$ de degré $\le q$, prédite par Poonen.

We give nonexistence results for rational points on the affine Drinfeld modular curve $Y_1\left(\mathfrak{p}\right)$ with $\mathfrak{p}$ a prime ideal of ${\mathbb{F}}_q\left[T\right]$. This curve classifies Drinfeld modules of rank $2$ with a torsion point of order $\mathfrak{p}$. The first statement concerns points defined over quadratic extensions of ${\mathbb{F}}_q\left(T\right)$ for $\mathfrak{p}$ of degree $3$ and cubic extensions of ${\mathbb{F}}_q\left(T\right)$ for $\mathfrak{p}$ of degree $4$ and $q\ge 7$. The second statement is valid under a duality condition between Hecke algebra and Drinfeld modular forms, and concerns points over extensions of degree $\le q$ whenever $deg\mathfrak{p}$ is sufficiently large. As a consequence we derive, under the same condition, a uniform bound for the torsion of rank-$2$ Drinfeld modules defined over extensions of ${\mathbb{F}}_q\left(T\right)$ of degree $\le q$, as predicted by Poonen.