the units of a ring spectrum and a logarithmic cohomology operation recall that if r is a commutative ring then the set r r of invertible elements of r is naturally an abelian group under multiplication this construction is a functor from commutative rings to abelian groups in general there is no obvious relation between the additive group of a ring r and the multiplicative group of units r however under certain circumstances one can define a homomorphism from a subgroup of r to a suitable completion of r e g the natural logarithm q 0 r or the p adic logarithm 1 pzp zp the logarithmic cohomology operation is a homotopy theoretic analogue of the above where r is a commutative s algebra and completion is bousfield localization with respect to a morava k theory the purpose of this paper is to give a formula for the logarithmic operation in certain contexts in terms of power operations before giving our results we briefly explain some of the concepts involved

sklyanin algebras and hilbert schemes of points we construct projective moduli spaces for torsion free sheaves on noncommutative projective planes these moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual commutative projective plane p2 the generic noncommutative plane corresponds to the sklyanin algebra s skl e constructed from an automorphism of infinite order on an elliptic curve e p2 in this case the fine moduli space of line bundles over s with first chern class zero and euler characteristic 1 n provides a symplectic variety that is a deformation of the hilbert scheme of n points on p2 e

blowup subalgebras of the sklyanin algebra we describe some interesting graded rings which are generated by degree 3 elements inside the sklyanin algebra s and prove that they have many good properties geometrically these rings r correspond to blowups of the sklyanin p2 at 7 or fewer points we show that the rings r are exactly those degree 3 generated subrings of s which are maximal orders in the quotient ring of the 3 veronese of s

a class of noncommutative projective surfaces let a k a 1 a 2 be a connected graded noetherian k algebra that is generated in degree one over an algebraically closed field k suppose that the graded quotient ring q a has the form q a k y t t 1 sigma where sigma is an automorphism of the integral projective surface y then we prove that a can be written as a naive blowup algebra of a projective surface x birational to y this enables one to obtain a deep understanding of the structure of these algebras for example generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme this is despite the fact that the simple objects in the quotient category qgr a will always be in 1 1 correspondence with the closed points of the scheme x

frobenius amplitude and strong vanishing theorems for vector bundles the primary goal of this paper is to systematically exploit the method of deligne illusie to obtain kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties the key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the frobenius or f amplitude the f amplitude enters into the statement of the basic vanishing theorem and this leads to the problem of calculating or at least estimating this number most of the work in this paper is devoted to doing this various situations

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

buildings elliptic curves and the k 2 local sphere we investigate a dense subgroup of the second morava stabilizer group given by a certain group of quasi isogenies of a supersingular elliptic curve in characteristic p the group acts on the bruhat tits building for gl2 through its action on the adic tate module this action has finite stabilizers giving a small resolution for the homotopy fixed point spectrum eh 2 hgal by spectra of topological modular forms here e2 is a version of morava e theory and gal gal

on the center of 3 dimensional and 4 dimensional sklyanin algebras abstract in this article a new proof is given of the description of the center of quadratic sklyanin algebras of global dimension three and four and the center of cubic sklyanin algebras of global dimension three the representation theory of the heisenberg groups h 2 h 3 and h 4 will play an important role in addition a new proof is given of van den bergh s result regarding noncommutative quadrics

simultaneous kummer congruences and mathbb e _ infty e orientations of ko and tmf building on results of ando hopkins and rezk we show the existence of uncountably many mathbb e infty string orientations of real k theory ko and of topological modular forms tmf generalizing the hat a resp the witten genus furthermore the obstruction to lifting an mathbb e infty string orientations from ko to tmf is identified with a classical iwasawa theoretic condition the common key to all these results is a precise understanding of the classical kummer congruences imposed for all primes simultaneously this result is of independent arithmetic interest

realizability of algebraic galois extensions by strictly commutative ring spectra we discuss some of the basic ideas of galois theory for commutative s algebras originally formulated by john rognes we restrict our attention to the case of finite galois groups and to global galois extensions we describe parts of the general framework developed by rognes central roles are played by the notion of strong duality and a trace mapping constructed by greenlees and may in the context of generalized tate cohomology we give some examples where algebraic data on coefficient rings ensures strong topological consequences we consider the issue of passage from algebraic galois extensions to topological ones by applying obstruction theories of robinson and goerss hopkins to produce topological models for algebraic galois extensions and the necessary morphisms of commutative s algebras examples such as the complex k theory spectrum as a ko algebra indicate that more exotic phenomena occur in the topological setting we show how in certain cases topological abelian galois extensions are classified by the same harrison groups as algebraic ones and this leads to computable harrison groups for such spectra we end by proving an analogue of hilbert s theorem 90 for the units associated with a galois extension

finiteness results for modular curves of genus at least 2 a curve x over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is modular if it is dominated by x 1 n for some n if in addition the image of its jacobian in j 1 n is contained in the new subvariety of j 1 n then x is called a new modular curve we prove that for each g 2 the set of new modular curves over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of genus g is finite and computable for the computability result we prove an algorithmic version of the de franchis severi theorem similar finiteness results are proved for new modular curves of bounded gonality for new modular curves whose jacobian is a quotient of j 0 n new with n divisible by a prescribed prime and for modular curves new or not with levels in a restricted set we study new modular hyperelliptic curves in detail in particular we find all new modular curves of genus 2 explicitly and construct what might be the complete list of all new modular hyperelliptic curves of all genera finally we prove that for each field k of characteristic zero and g 2 the set of genus g curves over k dominated by a fermat curve is finite and computable

some noncommutative minimal surfaces abstract in the ongoing programme to classify noncommutative projective surfaces connected graded noetherian domains of gelfand kirillov dimension three a natural question is to determine the minimal models within any birational class in this paper we show that the generic noncommutative projective plane corresponding to the three dimensional sklyanin algebra as well as noncommutative analogues of p 1 p 1 and the more general van den bergh quadrics satisfy very strong minimality conditions translated into an algebraic question where one is interested in a maximality condition we prove the following result theorem a let r be a sklyanin algebra or a van den bergh quadric that is infinite dimensional over its centre and let a r be any connected graded noetherian maximal order with the same graded quotient ring as r then up to taking veronese rings a is isomorphic to r let t be an elliptic algebra that is the coordinate ring of a noncommutative surface containing an elliptic curve then under an appropriate homological condition we prove that every connected graded noetherian overring of t is obtained by blowing down finitely many lines line modules of self intersection 1

residue complex for sklyanin algebras of dimension three we describe the residue complex for three dimensional sklyanin algebras which are the interesting special cases of quantum polynomial rings in three variables in particular we show that the multiplicities of the point modules in the residue complex are all one just as in the classical case of commutative polynomial rings in three variables we explain why the residue complex in the quantum case has the same multiplicities of point modules as that in the commutative case even if there are fewer point modules in the quantum case by pointing out two quantum anomalies

topological modular forms with level structure the cohomology theory known as mathrm tmf for topological modular forms is a universal object mapping out to elliptic cohomology theories and its coefficient ring is closely connected to the classical ring of modular forms we extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them along the way we produce a natural way to restrict to the cusps providing multiplicative maps from mathrm tmf with level structure to forms of k theory in particular this allows us to construct a connective spectrum mathrm tmf 0 3 consistent with properties suggested by mahowald and rezk this is accomplished using the machinery of logarithmic structures we construct a presheaf of locally even periodic elliptic cohomology theories equipped with highly structured multiplication on the log etale site of the moduli of elliptic curves evaluating this presheaf on modular curves produces mathrm tmf with level structure

k 1 local topological modular forms we construct the witten orientation of the topological modular forms spectrum tmf in the k 1 local setting by attaching e cells to the bordism theory mo we also identify the ko homology of tmf with the ring of divided congruences

irreducible representations of the 4 dimensional sklyanin algebra at points of infinite order abstract in 1982 sklyanin funct anal appl 16 1982 27 34 defined a family of graded algebras a e depending on an elliptic curve e and a point e which is not 4 torsion basic properties of these algebras were established in smith and stafford compositio math 83 1992 259 289 and a study of their representation theory was begun in levasseur and smith bull soc math france121 1993 35 90 the present paper classifies the finite dimensional simple a modules when is a point of infinite order sklyanin funct anal appl 17 1983 273 284 defines for each k n a representation of a in a certain k dimensional subspace of theta functions of order 2 k 1 we prove that these are irreducible representations and that any other simple module is obtained by twisting one of these by an automorphism of a the automorphism group of a is explicitly computed the method of proof relies on results in levasseur and smith in particular it is proved that every finite dimensional simple module is a quotient of a line module an important part of the analysis is a determination of the 1 critical a modules and the fact that such a module is equivalent to a quotient of a line module by a shifted line module

noncommutative graded domains with quadratic growth letk be an algebraically closed field and letr be a finitely generated connected gradedk algebra which is a domain of gelfand kirillov dimension two write the graded quotient ringq r ofr asd z z 1 for some automorphism of the division ringd we prove thatd is a finitely generated field extension ofk of transcendence degree one moreover we describer in terms of geometric data ifr is generated in degree one then up to a finite dimensional vector space r is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curvey this implies in particular thatr is noetherian thatr is primitive when and thatr is a finite module over its centre when ifr is not generated in degree one thenr will still be noetherian and primitive if has infinite order butr need not be noetherian when has finite order

naive noncommutative blowing up let b x mathscr l be the twisted homogeneous coordinate ring of an irreducible variety x over an algebraically closed field k with dim x 2 assume that c in x and in aut x are in sufficiently general position we show that if one follows the commutative prescription for blowing up x at c but in this noncommutative setting one obtains a noncommutative ring r r x c mathscr l with surprising properties 1 r is always noetherian but never strongly noetherian 2 if r is generated in degree one then the images of the r point modules in qgr r are naturally in one to one correspondence with the closed points of x however in both qgr r and gr r the r point modules are not parametrized by a projective scheme 3 while qgr r has finite cohomological dimension dim k h 1 mathscr o

orbifold genera product formulas and power operations we generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h using h tuples rather than pairs of commuting elements we show that our genera are in fact orbifold invariants and we prove integrality results for them if the genus arises from an h8 map into the morava lubin tate theory eh then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold m in terms of the genus of m itself our formula is the p typical analogue of the dijkgraaf moore verlinde verlinde formula for the orbifold elliptic genus r dijkgraaf et al elliptic genera of symmetric products and second quantized strings comm math phys 185 1 1997 197 209 it depends only on h and not on the genus

the sigma orientation is an h map in an earlier paper the authors constructed a natural map called the sigma orientation from the thom spectrum mu 6 to any elliptic spectrum mu 6 is an h ring spectrum and in this paper we show that if e c t is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p 0 then the sigma orientation is a map of h ring spectra

power operations in elliptic cohomology and representations of loop groups the first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve the second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups the third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part