Two theorems about maximal Cohen--Macaulay modules

This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely many isomorphism classes of maximal Cohen--Macaulay modules exist having ranks up to the sum of the ranks of $M$ and $N$. This has several corollaries. In particular it proves that a Cohen--Macaulay local ring of finite Cohen--Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen--Macaulay local ring of finite Cohen--Macaulay type is again of finite Cohen--Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic $p$ and dimension $d$ is $F$-rational if and only if the number of copies of $R$ splitting out of $R^{1/p^e}$ divided by $p^{de}$ has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the $F$-signature of the ring and give some of its properties.Comment: 14 page

Higher congruence companion forms

For a rational prime p � 3 we consider p-ordinary, Hilbert modular newforms f of weight k � 2 with associated p-adic Galois representations �f and mod pn reductions �f;n. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of �f;n to decomposition groups above p split then f has a companion form g modulo pn (in the sense that �f;n � �g;n �

Higher congruence companion forms

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $\rho_f$ and $\mod{p^n}$ reductions $\rho_{f,n}$. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of $\rho_{f,n}$ to decomposition groups above $p$ split then $f$ has a companion form $g$ modulo $p^n$ (in the sense that $\rho_{f,n}\sim \rho_{g,n}\otimes\chi^{k-1}$).Comment: 13 page

A Note on Transfinite M Theory and the Fine Structure Constant

In this short note, using notions from $p$-Adic QFT and $p$-branes, we derive the transfinite M $theoretical$ corrections $(\alpha_M)^{-1} = 100 + 61 \phi$ to El Naschie's inverse fine structure constant value $(\alpha_{HS})^{-1}= 100 + 60\phi$ which was based on a transfinite Heterotic string theory ormalism . $\phi$ is the Golden Mean $0.6180339...$. Our results are consistent with recent Astrophysical observations of he Boomerang and Maxima experiments, with previous results based on the four dimensional gravitational conformal anomaly calculations and with an enhanced hierarchy of the number of lines on Del Pezzo surfaces.Comment: 11 pages,LaTe

A graph polynomial for independent sets of bipartite graphs

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings and the number of perfect matchings. Most importantly, for bipartite graphs the polynomial encodes the number of independent sets (#BIS). We analyze the complexity of exact evaluation of the polynomial at rational points and show that for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial. We conjecture that a natural Markov chain can be used to approximately evaluate the polynomial for a range of parameters. The conjecture, if true, would imply an approximate counting algorithm for #BIS, a problem shown, by [Dyer et al. 2004], to be complete (with respect to, so called, AP-reductions) for a rich logically defined sub-class of #P. We give a mild support for our conjecture by proving that the Markov chain is rapidly mixing on trees. As a by-product we show that the "single bond flip" Markov chain for the random cluster model is rapidly mixing on constant tree-width graphs

Multiplicities Associated to Graded Families of Ideals

We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings, with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustata, and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We also prove an asymptotic "additivity formula" for limits of multiplicities, and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov, for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.Comment: 20 pages. The statement of Theorem 6.1 is corrected by adding the assumption that all ideals considered are nonzero. arXiv admin note: text overlap with arXiv:1301.561