Let $n,d,k\geq2,b,y$ and $\ell\geq3$ be positive integers with the greatest prime factor of b not exceeding k. It is proved that the equation $n (n+d) \dotsb (n+(k-1)d)=b y^{\ell}$ has no solution if d exceeds d1, where d1 equals 30 if $\ell =3$; 950 if $\ell =4$; $5\times 10^4$ if $\ell=5$ or 6; 108 if $\ell=7$, 8, 9 or 10; 1015 if $\ell \geq 11$. This confirms a conjecture of Erdős on the above equation for a large number of values of d.

We give a criterion to check if, given a prime power pr with r > 1, the only rational points of the modular curve X0+ (pr) are trivial (i.e. cusps or points furnished by complex multiplication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit (p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves.

Let k be a number field with ring of integers $\mathfrak{O}_k$, and let $\Gamma=A_4$ be the tetrahedral group. For each tame Galois extension N/k with group isomorphic to $\Gamma$, the ring of integers $\mathfrak{O}_{N}$ of N determines a class in the locally free class group $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$. We show that the set of classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ realized in this way is the kernel of the augmentation homomorphism from $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ to the ideal class group $\mathrm{Cl}(\mathfrak{O}_k)$. This refines a result of Godin and Sodaïgui (J. Number Theory98 (2003), 320–328) on Galois module structure over a maximal order in $k[\Gamma]$. To the best of our knowledge, our result gives the first case where the set of realizable classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ has been determined for a nonabelian group $\Gamma$ and an arbitrary number field k.

Let $\tau$ be a point in the upper half-plane such that the elliptic curve corresponding to $\tau$ can be defined over $\mathbb{Q}$, and let f be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f, we obtain a family of modular forms Dlf. In this paper we study the behavior of $D^l(f)(\tau)$ modulo the powers of a prime p > 3. We show that for $p \equiv 1 \bmod 3$ the quantities $D^l(f)(\tau)$, suitably normalized, satisfy Kummer-type congruences, and that for $p \equiv 2 \bmod 3$ the p-adic valuations of $D^l(f)(\tau)$ grow arbitrarily large. We prove these congruences by making a connection with a certain elliptic curve whose reduction modulo p is ordinary if $p \equiv 1 \bmod 3$ and supersingular otherwise.

We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.

We prove that, near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of ‘evenly spaced’ annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

A differential algebra of finite type over a field $\mathbb{k}$ is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated $\mathbb{k}$-algebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when $\text{char}\mathbb{k} = 0$. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse A-modules, and show how they are related to the Auslander property of the rigid dualizing complex of A.

We define invariants of the blow-Nash equivalence of Nash function germs, in a similar way to the motivic zeta functions of Denef and Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, to a generalized Euler characteristic for projective constructible arc-symmetric sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariants, but also Nash invariants of arc-symmetric sets. Our zeta functions enable one to distinguish the blow-Nash equivalence classes of Brieskorn polynomials of two variables. We prove moreover that there are no moduli for the blow-Nash equivalence in the case of an algebraic family with isolated singularities.

We introduce an essentially new Grothendieck topology, the Weil-étale topology, on schemes over finite fields. The cohomology groups associated with this topology should behave better than the standard étale cohomology groups. In particular there is a very natural definition of an Euler characteristic and a plausible conjecture relating the Euler characteristic of Z to the value of the zeta-function at s = 0. This conjecture is proved in certain cases.

Let X be an irreducible projective variety of dimension n in a projective space and let x be a point of X. Denote by Curvesd(X, x) the space of curves of degree d lying on X and passing through x. We will show that the number of components of Curvesd(X, x) for any smooth point x outside a subvariety of codimension $\geq 2$ is bounded by a number depending only on n and d. An effective bound is given. A key ingredient of the proof is an argument from Ein, Küchle and Lazarsfeld's work on Seshadri numbers.

We present an explicit formula for computing toric residues of ample divisors as a quotient of two determinants, à la Macaulay, where the numerator is a minor of the denominator. We present a combinatorial construction of a specific element of residue 1. We also give an irreducible representation of toric residues by extending the theory of subresultants to monomials of critical degree in the homogeneous coordinate ring of the corresponding toric variety.

Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese map and the Clifford torus.

For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K-theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$. They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of $G(\mathbb{C}[[{\rm t}]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of $G(\mathbb{C}[[{\rm t}]])$-modules.

For cohomological (respectively homological) coefficient systems ${\mathcal F}$ (respectively ${\mathcal V}$) on affine buildings X with Coxeter data of type $\widetilde{A}_d$, we give for any $k\ge1$ a sufficient local criterion which implies $H^k(X,{\mathcal F})=0$ (respectively $H_k(X,{\mathcal V})=0$). Using this criterion we prove a conjecture of de Shalit on the acyclicity of coefficient systems attached to hyperplane arrangements on the Bruhat–Tits building of the general linear group over a local field. We also generalize an acyclicity theorem of Schneider and Stuhler on coefficient systems attached to representations.

This paper deals with singularities of nonconfluent hypergeometric functions in several complex variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of the amoebas and the Newton polytopes of their defining polynomials. In particular, we show that the amoebas of classical discriminantal hypersurfaces are solid, that is, they possess the minimal number of complement components.