Let $G$ be a $p$ -adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$ , the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$ , into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$

$\ell$

$(x,y,\ell )$

$y=0$

$\ell <\exp (3^{k})$

We apply the endoscopic classification of automorphic forms on $U(3)$ to study the growth of the first Betti number of congruence covers of a Picard modular surface. As a consequence, we establish a case of a conjecture of Sarnak and Xue on cohomology growth.

We study primitive theta functions, which were first introduced by Shintani, in a purely local setting. We investigate a metaplectic representation of U(1) acting on the space of local primitive theta functions and give its explicit irreducible decomposition. As a by-product, we give a new proof of epsilon dichotomy for (U(1),U}(1)).

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.

In this paper we study the following question: Is it true that a generic hypersurface X of degree d in P$^n+1$, where (d,n)≠ (3,1), (2,2), (3,2), does not admit a non-trivial, non-isomorphic surjective map to another smooth variety Y, except of course P$^n$? It is easy to see that it is true for n=1,2. We try to prove this for n=3 and can exclude all possibilities for Y except Y= G(1,4)∩ P$^6$and Y=V$_22^s$, a special Fano threefold of type V$_22$ found by Mukai and Umemura in [MU].

We present some basic identities for hypergeometric functions associated with the integrals of Euler type. We give a geometrical proof for formulae such as the identity between the single and double integrals expressing Appell's hypergeometric series $F_1 (a, b, b^{\prime} c; x, y)$.

We prove $R = {\mathbf T}$ theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras ${\mathbf T}$ are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite flat group schemes ${\mathscr G}$ and ${\mathscr H}$ of order p over an arbitrarily tamely ramified discrete valuation ring admit an extension not killed by p.

In this paper, we give an explicit computable algorithm for the Zelevinsky–Aubert duals of irreducible representations of $p$-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations.

Fix an integer $n\geqslant 2$. To each non-zero point $\mathbf{u}$ in $\mathbb{R}^{n}$, one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$. To avoid trivialities, one restricts to points $\mathbf{u}$ whose coordinates are linearly independent over $\mathbb{Q}$. The resulting set of values is called the spectrum of these exponents. We show that, in an appropriate setting, any such spectrum is a compact connected set. In the case $n=3$, we prove moreover that it is a semi-algebraic set closed under component-wise minimum. For $n=3$, we also obtain a description of the spectrum of the exponents $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{2},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{3})$ recently introduced by Schmidt and Summerer.

T. Fukui introduced an invariant for the blow-analytic equivalence of real singularities. For a nondegenerate analytic function (germ) f, he discovered a formula for computing the one-dimensional invariant, denoted by A(f) := A1(f). We find a formula for A(f) for any f (real or complex, degenerate or not). We then define, and characterise, various notions of stability of A(f), using the formula. For real analytic f, the Fukui invariant with sign is defined, and computed by a similar formula. In the case where f is an analytic function of two complex variables, A(f) can also be computed using the tree-model of f.

We prove the Kneser-Tits conjecture for groups of index $^2E^{29}_{6,1}$ using an argument inspired by a 1968 paper by Veldkamp. We also prove that these groups are stably rational varieties.

We show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over a Clements–Lindström ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. This is an analogue of Hartshorne’s theorem that Grothendieck’s Hilbert scheme is connected. We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.

An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.

We define a complex horospherical transform on an affine symmetric space $X=G/H$ of Hermitian type and show that it has no kernel on the representations of the $H$-spherical holomorphic discrete series.

A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf ℱ on X, there exists an integer m0 such that m≥m0 implies Hi (X,ℱ⊗𝒪(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

We formulate and prove necessary and sufficient conditions for the Galois correspondence to hold for the ring of p-adic periods B+dR.

We give a generalization of Poitou–Tate duality to schemes of finite type over rings of integers of global fields.