We prove that the Hilbert–Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.

In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur’s question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel–Jacobi map admits a distinguished model over the rationals.

Let E/ℚ be an elliptic curve and p a prime of supersingular reduction for E. Denote by the anticyclotomic ℤp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/ℚ, we prove that has trivial Λ-corank and, in the process, also show that and both have Λ-corank two.

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.

In this paper, we develop a new necessary and sufficient condition for the vanishing of $4$-Massey products of elements in the modulo-$2$ Galois cohomology of a field. This new description allows us to define a splitting variety for $4$-Massey products, which is shown in the appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such $4$-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.

We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

Let ${\rm F}$ be a non-Archimedean locally compact field of residue characteristic $p$, let ${\rm D}$ be a finite-dimensional central division ${\rm F}$-algebra and let ${\rm R}$ be an algebraically closed field of characteristic different from $p$. We define banal irreducible ${\rm R}$-representations of the group ${\rm G}={\rm GL}_{m}({\rm D})$. This notion involves a condition on the cuspidal support of the representation depending on the characteristic of ${\rm R}$. When this characteristic is banal with respect to ${\rm G}$, in particular when ${\rm R}$ is the field of complex numbers, any irreducible ${\rm R}$-representation of ${\rm G}$ is banal. In this article, we give a classification of all banal irreducible ${\rm R}$-representations of ${\rm G}$ in terms of certain multisegments, called banal. When ${\rm R}$ is the field of complex numbers, our method provides a new proof, entirely local, of Tadić’s classification of irreducible complex smooth representations of ${\rm G}$.

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

In this article the degree of the discriminant of an elliptic pencil on a projective curve is upper-bounded by using the degree of its conductor and the genus of the base curve. This is done in the most general case, extending a method and a result of Szpiro (1981 and 1990a) and a result of Hindry and Silvermann. The difficult part, dealing with characteristic 2 and 3 and additive reductions, need the introduction of a new object – which we called ’conducteur efficace‘ – defined by using differentials and interestingly comparable to the usual conductor. This article ends with a few results in the arithmetical case – case corresponding to an inequality conjectured by the second author in 1978: (1) the proof of this inequality in the potentially good reduction cases; (2) the passage from the semi-stable reduction to the general case for a strong inequality.

We state certain product formulae for Jackson integrals associated with irreducible reduced root systems. The Jackson integral is defined here as a sum over any full-rank sublattice of the coweight lattice for the root system. In particular, a Weyl group symmetry classification of the Jackson integrals is done when they have an expression of a product of the Jacobi elliptic theta functions. Most of the product formulae investigated by Aomoto, Macdonald and Gustafson appear in the list of classifications. A new product formula for an F4 root system is included in it.

We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a ’hypergeometric sensitivity analysis‘ for small variations of constraint constants with creation operators and b-functions. In the second part, we study the indicial polynomial (b-function) along the hyperplane xi=0 via a correspondence between the optimal value of an integer programming problem and the roots of the indicial polynomial. Gröbner bases are used to prove theorems and give counter examples.

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.

We construct natural equivalences between derived categories of coherent sheaves on the local models for stratified Mukai and Atiyah flops (of type A).

By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic $K$-theory. We give a new and direct proof of Suslin’s result based on an exact sequence of categories of perfect modules. In fact, we prove a more general descent result for a pullback square of ring spectra and any localizing invariant. Our descent theorem contains not only Suslin’s result, but also Nisnevich descent of algebraic $K$-theory for affine schemes as special cases. Moreover, the role of the Tor-unitality condition becomes very transparent.

In this paper, constructions of Jordan algebras over commutative rings are given which place, within a general set-up, the classical Tits constructions of exceptional central simple Jordan algebras over fields. These are used to exhibit nontrivial Jordan algebra bundles over the affine plane with a given exceptional Jordan division algebra over k as the fibre. The associated principal F$_4$ bundles are shown to admit no reduction of the structure group to any proper connected reductive subgroup.

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

We associate to a group-like monoidal grupoid ${\mathcal C}$a principal bundle E satisfying most of the axioms defining a biextension. The obstruction to the existence of a genuine biextension structure on E is exhibited. When this obstruction vanishes, the biextension E is alternating and a trivialization of E induces a trivialization of${\mathcal C}$. The analogous theory for monoidal n-categories is also examined, as well as the appropriate generalization of these constructions in a sheaf-theoretic context. In the n-categorial situation, this produces a higher commutator calculus, in which some interesting generalizations of the notion of an alternating biextension occur. For n=2, the corresponding cocycles are constructed explicitly, by a partial symmetrization process, from the cocycle describing the n-category.

Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.