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In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the $\mod p$ cohomology of congruence subgroups of ${\rm SL}\sb n(\mathbb {Z})$ to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case $n=3$ in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations that are in no obvious way related to lower-dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.

In this note we study the geometry of torsors under flat and finite commutative group schemes of rank p above curves in characteristic p and above relative curves over a complete discrete valuation ring of inequal characteristics. In bothe cases we study the Galois action of the Galois group of the base field on these torsors. We also study degeneration of $\mu_p$-torsors from characteristic 0 to characteristic p and show that this degeneration is compatible with the Galois action. We then discuss the lifting of torsors under flat and commutative group schemes of rank p from positive to 0 characteristics. finally, for a proper and smooth curve X over a complete discrete valuation field of inequal characteristics we show the existence of a canonical Galois equivariant filtration on the first etale cohomology group of the geometric fibre of X with values in $\mu_p$. This apaper is the first of a series of papers [Sa] and [Sa-1] where we compute the vanishing cycles arising from degeneration of $\mu_p$-torsors above curves as well as the semi-stable reduction of these torsors and the Galois action on them., Comment: 17 pages amstex, weakend hypothesis in 2.4 the statement with the original hypothesis being wrong

This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered product of regular trees. We construct explicit arithmetic examples using quaternion algebras over totally real fields. Here we reduce the Ramanujan property to special cases of the Ramanujan-Petersson conjecture for holomorphic Hilbert modular forms, many of which are known.

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of G\"ollnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts $\not\equiv 0, \pm i$ ($mod$ 6), and a set of three theorems involving partitions into parts $\not\equiv 0, \pm i$ ($mod$ 7), $i=1,2,3$., Comment: 31 pages, 4 figures

In Proposition I of "Memoire sur les conditions de resolubilite des equations par radicaux", Galois established that any intermediate extension of the splitting field of a polynomial with rational coefficients is the fixed field of its galois group. We first state and prove the (dual) categorical interpretation of of this statement, which is a theorem about atomic sites with a representable point. In the general case, the point determines a proobject and it becomes (tautologically) prorepresentable. We state and prove the, mutatus mutatis, prorepresentable version of Galois theorem. In this case the classical group of automorphisms has to be replaced by the localic group of automorphisms. These developments form the content of a theory that we call "Localic Galois Theory". An straightforward corollary of this theory is the theorem: "A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point". This theorem was first proved in print in Joyal A, Tierney M. "An extension of the Galois Theory of Grothendieck", Mem. AMS 151, Theorem 1, Section 3, Chapter VIII. Our proof is completely independent of descent theory and of any other result in that paper., Comment: These results were presented at Como CT2000

In this paper we obtain a set of polynomials which are orthogonal with respect to the classical discrete weight function of the Charlier polynomials at which an extra point mass at x=0 is added. We construct a difference operator of infinite order for which these new discrete orthogonal polynomials are eigenfunctions., 11 pages

In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group $\Gamma_1(l)$. In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that $L(f,1) \not = 0$. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols., Comment: 14 pp., 1 figure, AMS-LaTeX

We study the equations of abelian surfaces embedded in P^{n-1} with a line bundle of polarization of type (1,n). For n>9, we show that the ideal of a general abelian surface with this polarization is generated by quadrics, and if the embedding is Heisenberg invariant, we give a very specific description of these quadrics. These quadrics are produced using a generalization of the Moore matrices which yields a uniform description of these ideals. We prove that these equations generate the ideals using a degeneration argument, degenerating abelian surfaces to schemes obtained from Stanley-Reisner ideals of certain triangulations of the torus. Furthermore, we obtain information on how to compute the moduli spaces of abelian surfaces of type (1,n) with canonical level structure. In a sequel to this paper, we will study the structure of the equations for n, 50 pages with 1 figure, author-supplied PostScript file available at http://www.math.columbia.edu/~psorin/eprints/abel-eq.ps (Final version) Plain TeX with epsf.tex and amssym.tex

The purpose of this paper is to construct a crepant resolution of quotient singularities by trihedral groups ( finite subgroups of SL(3,C) of certain type ), and prove that each Euler number of the minimal model is equal to the number of conjugacy classes. Trihedral singularities are 3-dimensional version of D_n-singularities, and they are non-isolated and many of them are not complete intersections. The resolution is similar to one of D_n-singularities. There is a nice combination of the toric resolution and Calabi-Yau resolution., 10 pages, Ams-Tex version 2.1, UTMS/94-18 (Dept. of Mathematical Science, Univ. of Tokyo)

In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite von Neumann algebra. Moreover, we compute the Brown measure of all unbounded R-diagonal operators in this class. As a particular case, we determine the Brown measure of z=xy^{-1}, where (x,y) is a circular system in the sense of Voiculescu, and we prove that for all positive integers n, z^n is in L^p(M) iff 0, Comment: 50 pages

In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame., 18 pages

The present paper is motivated by the observation that Weyl's equidistribution theorem for real sequences on a bounded interval can be formulated in a way which is also meaningful for sequences of selfadjoint operators on a Hilbert space. We shall provide general results on weak convergence of operator measures which yield this version of Weyl's theorem as a corollary. Further, by combining the above results with the von Neumann ergodic theorem, we will obtain a Cesaro convergence property, equivalently, an "ergodic theorem", which is valid for all (projectionvalued) spectral measures whose support is in a bounded interval, as well as for the more general class of positive operator-valued measures. Within the same circle of ideas we deduce a convergence property which completely characterizes those spectral measures associated with "strongly mixing" unitary transformations. The final sections are devoted to applications of the preceding results in the study of complexvalued Borel measures as well as to an extension of our results to summability methods other than CesAro convergence. In particular, we obtain a complete characterization, in purely measure theoretic terms, of those complex measures on a bounded interval whose Fourier-Stieltjes coefficients converge to zero.

We establish sharp asymptotically optimal strategies for the problem of online prediction with history dependent experts. The prediction problem is played (in part) over a discrete graph called the d dimensional de Bruijn graph, where d is the number of days of history used by the experts. Previous work Drenska and Kohn ( arXiv:2007.12732 , 2020) established $$O(\varepsilon )$$ optimal strategies for $$n=2$$ experts and $$d\le 4$$ days of history, while Drenska and Kohn (J Nonlinear Sci 30. 30(1), 137–173, 2020) established $$O(\varepsilon ^{1/3})$$ optimal strategies for all $$n\ge 2$$ and all $$d\ge 1$$ , where the game is played for N steps and $$\varepsilon =N^{-1/2}$$ . In this paper, we show that the optimality conditions over the de Bruijn graph correspond to a graph Poisson equation, and we establish $$O(\varepsilon )$$ optimal strategies for all values of n and d.

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $\rho$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,\rho)$ admits a Calder\'on--Zygmund decomposition which involves a particular family of sets, called Calder\'on--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid we shall prove a Fefferman--Stein type inequality, involving the dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009), 277--295]., Comment: Math. Z. (to appear)