Let $\varphi :A\to K\{\tau \}$ be a Drinfeld module of rank $2$ with generic characteristic, and suppose that the endomorphism ring of $\varphi$ induces a Drinfeld module $\psi :B\to K\{\tau \}$ of rank $1$. Let $a\in K$. We prove that the set of places $\wp$ of $K$ for which $a$ generates $\varphi ({\mathbb{𝔽}}_{\wp })$ as an $A$-module has a density. Furthermore, we show that this density is positive other than in some standard exceptional cases.

We also revisit Artin’s problem for Drinfeld modules of rank $1$, first considered by Hsu and Yu. A key point is that our methods do not require that $A$ be a principal ideal domain. We are also able to generalize a Brun–Titchmarsh theorem for function fields proved by Hsu.

We introduce the $G$-stable rank of a higher order tensors over perfect fields. The $G$-stable rank is related to the Hilbert–Mumford criterion for stability in geometric invariant theory. We will relate the $G$-stable rank to the tensor rank and slice rank. For numerical applications, we express the $G$-stable rank as a solution to an optimization problem. Over the field ${\mathbb{𝔽}}_3$ we discuss an application to the cap set problem.

Let $G$ be an infinite-dimensional representation-theoretic Kac–Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on $G$ induced from unramified cusp forms on finite-dimensional Levi subgroups of maximal parabolic subgroups. Under a natural condition on maximal parabolic subgroups, we prove the cuspidal Eisenstein series are entire on the full complex plane.

Let $I_1,\dots ,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J=I_1\cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.

We make a geometric study of the geometric rank of tensors recently introduced by Kopparty et al. Results include classification of tensors with degenerate geometric rank in ${ℂ}^3\otimes {ℂ}^3\otimes {ℂ}^3$, classification of tensors with geometric rank two, and showing that upper bounds on geometric rank imply lower bounds on tensor rank.

We show that the average size of the $2$-Selmer group of the family of Jacobians of nonhyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}∕2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava’s orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_6$, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.

This work concerns finite free complexes with finite-length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\text{dim} R$, which is the smallest possible value, and, in particular, on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. The key idea in the proof is the construction of a suitable Ulrich module, or, in the latter case, a sequence of modules that have the Ulrich property asymptotically, and with good convergence properties in the rational Grothendieck group of $R$. Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety.

We obtain an asymptotic formula for the mean value of $L$-functions associated to cubic characters over ${\mathbb{𝔽}}_q[T]$. We solve this problem in the non-Kummer setting when $q\equiv 2(\text{mod }3)$ and in the Kummer setting when $q\equiv 1(\text{mod }3)$. In the Kummer setting, the mean value over the complete family of cubic characters was never addressed in the literature (over number fields or function fields). The proofs rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the pioneer work of Kubota. In the non-Kummer setting, we display some explicit (and unexpected) cancellation between the main term and the dual term coming from the approximate functional equation of the $L$-functions. Exhibiting the cancellation involves evaluating sums of residues of a variant of the generating series of cubic Gauss sums.