This paper establishes a version of the Chebotarev density theorem in which number fields are replaced by 3-manifolds.

Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

In this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

For an abelian variety $A$ over a number field $k$ we discuss the maximal divisible subgroup of ${\mathrm{H} }^{1} (k, A)$ and its intersection with the subgroup Ш$(A/ k)$. The results are most complete for elliptic curves over $ \mathbb{Q} $.

Let X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

Let $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $. In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, $q\lt {q}_{ \mathsf{res} } $, for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $, ${C}_{n} $, ${D}_{n} $.

We consider the analogue of the André–Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André–Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.

In this work, we study the deformation theory of ${\mathcal {E}}_n$-rings and the ${\mathcal {E}}_n$ analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence $A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the ${\mathcal {E}}_n$-tangent complex and ${\mathcal {E}}_n$-Hochschild cohomology of an ${\mathcal {E}}_n$-ring $A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups, $B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here ${\mathfrak B}^n\!A$ is an enriched $(\infty ,n)$-category constructed from $A$, and ${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of ${\mathfrak B}^n\!A$. These groups are associated to moduli problems in ${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital ${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex $T_A[-n]$ is a nonunital ${\mathcal {E}}_{n+1}$-algebra. The ${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known ${\mathcal {E}}_{n+1}$-algebra structure on ${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed $n$-manifolds with coefficients given by ${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

Let $p\gt 2$ be prime, and let $F$ be a totally real field in which $p$ is unramified. We give a sufficient criterion for a $\mathrm{mod} \hspace{0.167em} p$ Galois representation to arise from a $\mathrm{mod} \hspace{0.167em} p$ Hilbert modular form of parallel weight one, by proving a ‘companion forms’ theorem in this case. The techniques used are a mixture of modularity lifting theorems and geometric methods. As an application, we show that Serre’s conjecture for $F$ implies Artin’s conjecture for totally odd two-dimensional representations over $F$.

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)$.

Let $k$ be a number field with algebraic closure $ \overline{k} $, and let $S$ be a finite set of primes of $k$ containing all the infinite ones. Let $E/ k$ be an elliptic curve, ${\mit{\Gamma} }_{0} $ be a finitely generated subgroup of $E( \overline{k} )$, and $\mit{\Gamma} \subseteq E( \overline{k} )$ the division group attached to ${\mit{\Gamma} }_{0} $. Fix an effective divisor $D$ of $E$ with support containing either: (i) at least two points whose difference is not torsion; or (ii) at least one point not in $\mit{\Gamma} $. We prove that the set of ‘integral division points on $E( \overline{k} )$’, i.e., the set of points of $\mit{\Gamma} $ which are $S$-integral on $E$ relative to $D, $ is finite. We also prove the ${ \mathbb{G} }_{\mathrm{m} } $-analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

In this paper we prove that surfaces of general type with irregularity $q\geq 3$ are rationally cohomologically rigidified, and so are minimal surfaces $S$ with $q(S)= 2$ unless ${ K}_{S}^{2} = 8\chi ({ \mathcal{O} }_{S} )$. Here a surface $S$ is said to be rationally cohomologically rigidified if its automorphism group $\mathrm{Aut} (S)$ acts faithfully on the cohomology ring ${H}^{\ast } (S, \mathbb{Q} )$. As examples we give a complete classification of surfaces isogenous to a product with $q(S)= 2$ that are not rationally cohomologically rigidified.

We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann  [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).

We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.

We provide an unconditional proof of the André–Oort conjecture for the coarse moduli space 𝒜2,1 of principally polarized abelian surfaces, following the strategy outlined by Pila–Zannier.

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces.

Let $K$ be a finitely generated extension of $\mathbb {Q}$. We consider the family of $\ell $-adic representations ($\ell $ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\ell $-adic cohomology of a separated scheme of finite type over $K$. We prove that the fields cut out from the algebraic closure of $K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

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 $A_5$.