We show that any toroidal DM stack $X$ with finite diagonalizable inertia possesses a maximal toroidal coarsening $X_{tcs}$ such that the morphism $X\to X_{tcs}$ is logarithmically smooth.

Further, we use torification results of Abramovich and Temkin (2017) to construct a destackification functor, a variant of the main result of Bergh (2017), on the category of such toroidal stacks $X$. Namely, we associate to $X$ a sequence of blowings up of toroidal stacks ${\widetilde{\mathcal{ℱ}}}_{X}Y\to X$ such that $Y_{tcs}$ coincides with the usual coarse moduli space $Y_{cs}$. In particular, this provides a toroidal resolution of the algebraic space $X_{cs}$.

Both $X_{tcs}$ and ${\widetilde{\mathcal{ℱ}}}_X$ are functorial with respect to strict inertia preserving morphisms $X^{\prime }\to X$.

Finally, we use coarsening morphisms to introduce a class of nonrepresentable birational modifications of toroidal stacks called Kummer blowings up.

These modifications, as well as our version of destackification, are used in our work on functorial toroidal resolution of singularities.

We give analogues of the Auslander correspondence for two classes of triangulated categories satisfying certain finiteness conditions. The first class is triangulated categories with additive generators and we consider their endomorphism algebras as the Auslander algebras. For the second one, we introduce the notion of $\left[1\right]$-additive generators and consider their graded endomorphism algebras as the Auslander algebras. We give a homological characterization of the Auslander algebras for each class. Along the way, we also show that the algebraic triangle structures on the homotopy categories are unique up to equivalence.

This article has three goals: First, we generalize the result of Deuring and Serre on the characterization of supersingular locus to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch–Kato conjecture.

We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen–Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological, categorical. We relate them to other classes of ideals and rings in the literature.

Pour une variété quasi-projective, lisse, géométriquement intègre sur un corps de nombres $k$, on montre que l’obstruction de descente itérée est équivalente à l’obstruction de descente. Ceci généralise un résultat de Skorobogatov, et ceci répond à une question ouverte de Poonen. Les outils principaux sont la notion de sous-groupe de Brauer invariant et la notion d’obstruction de Brauer–Manin étale invariante pour une $k$-variété munie d’une action d’un groupe linéaire connexe.

For a quasi-projective smooth geometrically integral variety over a number field $k$, we prove that the iterated descent obstruction is equivalent to the descent obstruction. This generalizes a result of Skorobogatov, and this answers an open question of Poonen. Our main tools are the notion of invariant Brauer subgroup and the notion of invariant étale Brauer–Manin obstruction for a $k$-variety equipped with an action of a connected linear algebraic group.

If $w$ is a word in $d>1$ letters and $G$ is a finite group, evaluation of $w$ on a uniformly randomly chosen $d$-tuple in $G$ gives a random variable with values in $G$, which may or may not be uniform. It is known that if $G$ ranges over finite simple groups of given root system and characteristic, a positive proportion of words $w$ give a distribution which approaches uniformity in the limit as $\left|G\right|\to \infty$. In this paper, we show that the proportion is in fact $1$.

We prove the conjecture of Yui and Zagier concerning the factorization of the resultants of minimal polynomials of Weber class invariants. The novelty of our approach is to systematically express differences of certain Weber functions as products of Borcherds products.

The main result of this paper is the following: for all $b\in \mathbb{Z}$ there exists $k=k\left(b\right)$ such that

$$max\left\{\left|A^{\left(k\right)}\right|,\left|{\left(A+u\right)}^{\left(k\right)}\right|\right\}\ge |A{|}^b,$$

for any finite $A\subset \mathbb{Q}$ and any nonzero $u\in \mathbb{Q}$. Here, $\left|A^{\left(k\right)}\right|$ denotes the $k$-fold product set $\left\{a_1\cdots a_k:a_1,\dots ,a_k\in A\right\}$.

Furthermore, our method of proof also gives the following $l_{\infty }$ sum-product estimate. For all $\gamma >0$ there exists a constant $C=C\left(\gamma \right)$ such that for any $A\subset \mathbb{Q}$ with $\left|AA\right|\le K\left|A\right|$ and any $c_1,c_2\in \mathbb{Q}\setminus \left\{0\right\}$, there are at most $K^C|A{|}^{\gamma }$ solutions to

$$c_{1}x+c_{2}y=1,\left(x,y\right)\in A\times A.$$

In particular, this result gives a strong bound when $K=|A{|}^{𝜖}$, provided that $𝜖>0$ is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem.

In further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products.

We utilize a query-complexity analogue of the polynomial Freiman–Ruzsa conjecture, due to Pälvölgyi and Zhelezov (2020). This new tool replaces the role of the complicated setup of Bourgain and Chang (2004), which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.

We study Heath-Brown’s and Serre’s dimension growth conjecture (proved by Salberger) when the degree $d$ grows. Recall that Salberger’s dimension growth results give bounds of the form $O_{X,𝜀}\left(B^{dimX+𝜀}\right)$ for the number of rational points of height at most $B$ on any integral subvariety $X$ of ${\mathbb{P}}_{\mathbb{Q}}^n$ of degree $d\ge 2$, where one can write $O_{d,n,𝜀}$ instead of $O_{X,𝜀}$ as soon as $d\ge 4$. We give the following simplified and strengthened forms of these results: we remove the factor $B^{𝜀}$ as soon as $d\ge 5$, we obtain polynomial dependence on $d$ of the implied constant, and we give a simplified, self-contained approach for $d\ge 16$. Along the way, we improve the well-known bounds due to Bombieri and Pila on the number of integral points of bounded height on affine curves and those by Walsh on the number of rational points of bounded height on projective curves. This leads to a slight sharpening of a recent estimate due to Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao on the size of the $2$-torsion subgroup of the class group of a degree $d$ number field. Our treatment builds on recent work by Salberger, who brings in many primes in Heath-Brown’s variant of the determinant method, and on recent work by Walsh and by Ellenberg and Venkatesh who bring in the size of the defining polynomial. We also obtain lower bounds showing that one cannot do better than polynomial dependence on $d$.