We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.

The numerical dimension is a numerical measure of the positivity of a pseudoeffective divisor $L$. There are several proposed definitions of the numerical dimension due to Nakayama and Boucksom et al. We prove the equality of these notions and give several additional characterizations. We also prove some new properties of the numerical dimension.

We prove a very general case of the parity conjecture for Selmer groups of elliptic curves over totally real fields, as well as slightly less general results for classical modular forms, Hilbert modular forms of parallel weight two and for abelian varieties with real multiplication.

Let $R$ be a commutative ring with $1$ and $q$ an invertible element of $R$. The (specialized) quantum group $U=U_q\left({\mathfrak{g}\mathfrak{l}}_n\right)$ over $R$ of the general linear group acts on mixed tensor space $V^{\otimes r}\otimes {V^{\ast }}^{\otimes s}$, where $V$ denotes the natural $U$-module $R^n$, $r$ and $s$ are nonnegative integers and $V^{\ast }$ is the dual $U$-module to $V$. The image of $U$ in ${End}_R\left(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s}\right)$ is called the rational $q$-Schur algebra $S_q\left(n;r,s\right)$. We construct a bideterminant basis of $S_q\left(n;r,s\right)$. There is an action of a $q$-deformation ${\mathfrak{B}}_{r,s}^n\left(q\right)$ of the walled Brauer algebra on mixed tensor space centralizing the action of $U$. We show that ${End}_{{\mathfrak{B}}_{r,s}^n\left(q\right)}\left(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s}\right)=S_q\left(n;r,s\right)$. By a previous result, the image of ${\mathfrak{B}}_{r,s}^n\left(q\right)$ in ${End}_R\left(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s}\right)$ is ${End}_U\left(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s}\right)$. Thus, a mixed tensor space as $\left(U,{\mathfrak{B}}_{r,s}^n\left(q\right)\right)$-bimodule satisfies Schur–Weyl duality.

Let $G_1$ and $G_2$ be absolutely almost simple algebraic groups of types $B_{\ell }$ and $C_{\ell }$, respectively, defined over a number field $K$. We determine when $G_1$ and $G_2$ have the same isomorphism or isogeny classes of maximal $K$-tori. This leads to the necessary and sufficient conditions for two Zariski-dense $S$-arithmetic subgroups of $G_1$ and $G_2$ to be weakly commensurable.

We consider models for genus-$1$ curves of degree $5$, which arise in explicit $5$-descent on elliptic curves. We prove a theorem on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and give an algorithm for computing such models. Finally we describe how to reduce genus-$1$ models of degree $5$ defined over $\mathbb{Q}$.

We study the number of nonzero coefficients of cyclotomic polynomials ${\Phi }_m$, where $m$ is the product of two distinct primes.

Let $f:{\mathbb{A}}^N\to {\mathbb{A}}^N$ be a regular polynomial automorphism defined over a number field $K$. For each place $v$ of $K$, we construct the $v$-adic Green functions $G_{f,v}$ and $G_{f^{-1},v}$ (i.e., the $v$-adic canonical height functions) for $f$ and $f^{-1}$. Next we introduce for $f$ the notion of good reduction at $v$, and using this notion, we show that the sum of $v$-adic Green functions over all $v$ gives rise to a canonical height function for $f$ that satisfies a Northcott-type finiteness property. Using an earlier result, we recover results on arithmetic properties of $f$-periodic points and non-$f$-periodic points. We also obtain an estimate of growth of heights under $f$ and $f^{-1}$, which was independently obtained by Lee by a different method.

Swinnerton-Dyer considered the proportion of twists of an elliptic curve with full $2$-torsion that have $2$-Selmer group of a particular dimension. Swinnerton-Dyer obtained asymptotic results on the number of such twists using an unusual notion of asymptotic density. We build on this work to obtain similar results on the density of twists with particular rank of $2$-Selmer group using the natural notion of density.