On the theta operator for modular forms modulo prime powers

We consider the classical theta operator $\theta$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $\theta$ mod $p^m$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least 2. Thus, the natural expectation that $\theta$ mod $p^m$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the $\theta$ operator on eigenforms mod $p^m$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the $\theta$-operator mod $p^m$ gives an explicit weight bound on the twist of a modular mod $p^m$ Galois representation by the cyclotomic character

On modular Galois representations modulo prime powers

We study modular Galois representations mod $p^m$. We show that there are three progressively weaker notions of modularity for a Galois representation mod $p^m$: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level $M$. Using results of Hida we display a `stripping-of-powers of $p$ away from the level' type of result: A mod $p^m$ strongly modular representation of some level $Np^r$ is always dc-weakly modular of level $N$ (here, $N$ is a natural number not divisible by $p$). We also study eigenforms mod $p^m$ corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod $p^m$ to any `dc-weak' eigenform, and hence to any eigenform mod $p^m$ in any of the three senses. We show that the three notions of modularity coincide when $m=1$ (as well as in other, particular cases), but not in general

Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

We present a new method for constructing $D$-dimensional minimally superintegrable systems based on block coordinate separation of variables. We give two new families of superintegrable systems with $N$ ($N\leq D$) singular terms of the partitioned coordinates and involving arbitrary functions. These Hamiltonians generalize the singular oscillator and Kepler systems. We derive their exact energy spectra via separation of variables. We also obtain the quadratic algebras satisfied by the integrals of motion of these models. We show how the quadratic symmetry algebras can be constructed by novel application of the gauge transformations from those of the non-partitioned cases. We demonstrate that these quadratic algebraic structures display an universal nature to the extent that their forms are independent of the functions in the singular potentials.Comment: 13 pages, no figure; Version to appear in Annals of Physic

Sequential importance sampling for multiway tables

We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates sampling values at each step to properties of the associated toric ideal using computational commutative algebra. In particular, the property of interval cell counts at each step is related to exponents on lead indeterminates of a lexicographic Gr\"{o}bner basis. Also, the approximation of integer programming by linear programming for sampling is related to initial terms of a toric ideal. We apply the algorithm to examples of contingency tables which appear in the social and medical sciences. The numerical results demonstrate that the theory is applicable and that the algorithm performs well.Comment: Published at http://dx.doi.org/10.1214/009053605000000822 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras

We present a useful proposition for discovering extended Laplace-Runge-Lentz vectors of certain quantum mechanical systems. We propose a new family of superintegrable systems and construct their integrals of motion. We solve these systems via separation of variables in spherical coordinates and obtain their exact energy eigenvalues and the corresponding eigenfunctions. We give the quadratic algebra relations satisfied by the integrals of motion. Remarkably these algebra relations involve the Casimir operators of certain higher rank Lie algebras in the structure constants.Comment: Latex 12 pages, no figure