Mathematics

Let X be a smooth projective quadric defined over a field of characteristic 2. We prove that in the Chow group of codimension 2 or 3 of X the torsion subgroup has at most two elements. In codimension 2, we determine precisely when this torsion subgroup is nontrivial. In codimension 3, we show that there is no torsion if { dimX??1 .} This extends the analogous results in characteristic different from 2, obtained by Karpenko in the nineteen-nineties.

In this article, we improve our main results from \emph{Chow groups and L -derivatives of automorphic motives for unitary groups} in two direction: First, we allow ramified places in the CM extension E/F at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla--Rapoport conjecture for exotic smooth Rapoport--Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures.

We give a full description of the functions F of degree 2 and conductor 1 in the general framework of the extended Selberg class. This is performed by means of a new numerical invariant χ F , which is easily computed from the data of the functional equation. We show that the value of χ F gives a precise description of the nature of F , thus providing a sharp form of the classical converse theorems of Hecke and Maass. In particular, our result confirms, in the special case under consideration, the conjecture that the functions in the Selberg class are automorphic L -functions.

We study quadratic polynomials giving bijections from the integer lattice points of sectors of R 2 onto N 0 , called packing polynomials. We determine all quadratic packing polynomials on rational sectors. This generalizes results of Stanton, Nathanson, and Fueter and Pólya.

The number of non-isomorphic cubic fields L sharing a common discriminant d(L) = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a multiplet M(d) = (L(1) ,..., L(m)). In this paper, the information in all existing tables of totally real cubic number fields L with positive discriminants d(L) < 10000000 is extended by computing the differential principal factorization types tau(L) in (alpha1, alpha2, alpha3, beta1, beta2, gamma, delta1, delta2, epsilon) of the members L of each multiplet M(d) of non-cyclic fields, a new kind of arithmetical invariants which provide succinct information about ambiguous principal ideals and capitulation in the normal closures N of non-Galois cubic fields L. The classification is arranged with respect to increasing 3-class rank of the quadratic subfields K of the S3-fields N, and to ascending number of prime divisors of the conductor f of N/K. The Scholz conjecture concerning the distinguished index of subfield units (U(N) : U(0)) = 1 for ramified extensions N/K with conductor f > 1 is refined and verified.

Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of p -adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their p -adic pro-étale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of ??-adic pro-étale cohomology, for ?��?p .

We study the cohomology of families of (?,?) -modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify rank- 1 (?,?) -modules and deduce that triangulations of pseudorigid families of (?,?) -modules can be interpolated, extending a result of [KPX14]. We then apply this to study extended eigenvarieties at the boundary of weight space, proving in particular that the eigencurve is proper at the boundary and that Galois representations attached to certain characteristic p points are trianguline.

We study the Selmer group associated to a p -ordinary newform f∈ S 2r ( Γ 0 (N)) over the anticyclotomic Z p -extension of an imaginary quadratic field K/Q . Under certain assumptions, we prove that this Selmer group has no proper Λ -submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants of congruent modular forms by the present authors.

We survey a number of different methods for computing L(?,1?�k) for a Dirichlet character ? , with particular emphasis on quadratic characters. The main conclusion is that when k is not too large (for instance k??00 ) the best method comes from the use of Eisenstein series of half-integral weight, while when k is large the best method is the use of the complete functional equation, unless the conductor of ? is really large, in which case the previous method again prevails.

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear q -difference equation with rational function coefficients. This Galois group encodes the possible polynomial differential relations among the solutions of the equation. We apply our results to compute the differential Galois groups of several concrete q -difference equations, including for the colored Jones polynomial of a certain knot.