We prove the parity conjecture for the ranks of p-power Selmer groups (p⁄=2) of a large class of elliptic curves defined over totally real number fields.

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin’s conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of motivic Euler product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.

Let R(n,θ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding nθ/2. When θ>e−1/3 a lower bound for R(n,θ) of the expected order of magnitude is established, and when θ>365/592, it is shown that R(n,θ)>0 holds for large n. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed θ>0. This last result is the key to bound R(n,θ) from below.

We prove that the Iitaka conjecture Cn,m for algebraic fibre spaces holds up to dimension six, that is, when n≤6.

We show an arithmetic generalization of the recent work of Lazarsfeld–Mustaţǎ which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.

Let 𝒱 be a complete discrete valuation ring of unequal characteristic with perfect residue field. Let be a separated smooth formal 𝒱-scheme, 𝒵 be a normal crossing divisor of , be the induced formal log-scheme over 𝒱 and be the canonical morphism. Let X and Z be the special fibers of and 𝒵, T be a divisor of X and ℰ be a log-isocrystal on overconvergent along T, that is, a coherent left -module, locally projective of finite type over . We check the relative duality isomorphism: . We prove the isomorphism , which implies their holonomicity as -modules. We obtain the canonical morphism ρℰ : uT,+(ℰ)→ℰ(†Z). When ℰ is moreover an isocrystal on overconvergent along T, we prove that ρℰ is an isomorphism.

For a Whitney stratification S of a space X (or, more generally, a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category EP≤2(X,S), called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors 2Funct(EP≤2(X,S),Cat) from the exit-path 2-category to the 2-category of small categories.

We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

This paper gives the classification of the Whittaker unitary dual for affine graded Hecke algebras of type E. By the Iwahori–Matsumoto involution, this is also equivalent to the classification of the spherical unitary dual for type E. Together with some results of Barbasch and Moy (D. Barbasch and A. Moy, Unitary spherical spectrum for p-adic classical groups, Acta Appl. Math. 44 (1996), 3–37; D. Barbasch, The spherical unitary spectrum of split classical real and p-adic groups, Preprint (2006), math/0609828) and Ciubotaru (D. Ciubotaru, The Iwahori spherical unitary dual of the split group of type F4, Represent. Theory 9 (2005), 94–137), this work completes the classification of the Whittaker Iwahori-spherical unitary dual or, equivalently, the spherical unitary dual of any split p-adic group.