We consider the polynomial representation S(V*) of the rational Cherednik algebra Hc(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and m∈ℕ the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous singular polynomials of degree d−1+hm, where h is the Coxeter number of W; these polynomials generate an Hc (W) submodule with the parameter c=(d−1)/h+m. We express these singular polynomials through the Saito polynomials which are flat coordinates of the Saito metric on the orbit space V/W. We also show that this exhausts all the singular polynomials in the isotypic component of the reflection representation V for any constant parameter c.

We prove Mazur and Rubin's $\Lambda$-adic Gross–Zagier conjecture (under some restrictive hypotheses), which relates Heegner points in towers of number fields to the 2-variable p-adic L-function. The result generalizes Perrin-Riou's p-adic Gross–Zagier theorem.

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

Desingularized fiber products of semi-stable elliptic surfaces with Hetale3=0 are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of arithmetic degenerations of rigid Calabi–Yau threefolds.

This is a study of the asymptotic behaviour of solutions of p-adic linear differential equations near the boundary of their convergence disks. We prove Dwork’s conjecture on the logarithmic growth of solutions in generic versus special disks.

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S(Γ)=Γ\X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup Γ of isometries of X) to restrict non-trivially to a compact locally symmetric subspace SH(Γ)=Δ\Y of Γ\X. The restriction is in a ‘virtual’ sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in $\Bbb C$n and $\Bbb C$m, then low degree cohomology classes on the variety S(Γ) restrict non-trivially to the subvariety SH(Γ); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S(Γ).

We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

A technical ingredient in Faltings’ original approach to $p$-adic comparison theorems involves the construction of $K({\it\pi},1)$-neighborhoods for a smooth scheme $X$ over a mixed characteristic discrete valuation ring with a perfect residue field: every point $x\in X$ has an open neighborhood $U$ whose generic fiber is a $K({\it\pi},1)$ scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in $p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma.

We prove a portion of a conjecture of Conrad, Diamond, and Taylor, yielding some new cases of the Fontaine–Mazur conjectures, specifically, the modularity of certain potentially Barsotti–Tate Galois representations. The proof follows the template of Wiles, Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered $\phi_1$-modules of Breuil.

In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$. Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry.

we find a minimal differential graded (dg) operad whose generic representations in ${\mathbb r}^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to ${\mathbb r}^n$ which satisfy the nijenhuis integrability condition. this operad is of a surprisingly simple origin: it is the cobar construction on the quadratic operad of homologically trivial dg lie algebras. as a by-product we obtain a strong-homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifolds.

It is proved that a product of four or more terms of positive integers in arithmetic progression with common difference a prime power is never a square. More general results are given which completely solve (1.1) with gcd(n, d)=1, k≥3 and 1<d≤104.

Let $K$ be a number field of degree $n$, and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$-functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$, respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$, $D_n$ for $n=3,4,5$, $S_4$ or $A_5$.

Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes $\cal S$ that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel–Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside $\cal S$, any prime that divides the algebraic part of the value at s=1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the ‘boundary primes’ in terms of expressions of the form $N_{F/{\mathbb{Q}}}(\epsilon^{k-1} - 1)$, where ε is a totally positive unit of F.

This paper analyzes the space HomH(π, 1), where π is an irreducible, tame supercuspidal representation of GL(n) over a p-adic field and H is a unitary group in n variables contained in GL(n). It is shown that this space of linear forms has dimension at most one. The representations π which admit nonzero H-invariant linear forms are characterized in several ways, for example, as the irreducible, tame supercuspidal representations which are quadratic base change lifts.

Given an intersection of two quadrics $X\subset { \mathbb{P} }^{m- 1} $, with $m\geq 9$, the quantitative arithmetic of the set $X( \mathbb{Q} )$ is investigated under the assumption that the singular locus of $X$ consists of a pair of conjugate singular points defined over $ \mathbb{Q} (i)$.

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan et al. [Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), 533–544], and Chernousov and Merkurjev [Essential dimension of spinor and Clifford groups, Algebra Number Theory 8 (2014), 457–472] to fields of characteristic different from two. We also complete the determination of generic stabilizers in spin and half-spin groups of low rank.