The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary  generalized local cohomology  modules.  Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$  are  finitely generated  $R$-modules and $d, t$ are two integers. We prove that $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)...

The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact. Most notably, the Birch and Swinnerton-Dyer conjecture, which relates the arithmetic of an elliptic curve to the analytic behaviour of its associated L-series, is still unproved in spite of important advances in the last decades, some of which are recalled in Chapter 1. In the 1960’s the pione...

Let π be a cuspidal, automorphic representation of GSp4 attached to a Siegel modular form of degree 2. We refine the method of Furusawa [9] to obtain an integral representation for the degree-8 L-function L(s, π × τ), where τ runs through certain cuspidal, automorphic representation of GL2. Our calculations include the case of square-free level for the p-adic components of τ , and a wide class ...

Let K/F be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified p-adic representation ρ of the absolute Galois group of K is determined (up to equivalence) by the characteristic polynomials of Frobenius elements Frv at any set of primes v of K of degree 1 over F . Here we prove an analogue for GL(n), nam...

Murphy’s Law for the Hilbert scheme (and the Chow variety, and moduli spaces of surfaces of general type, and stable maps, and nodal and cuspidal plane curves, and . . . )

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the LorentzMinkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, gener...

In this paper we prove that there is an infinite sequence of pairs of plane cuspidal curves Cm,1 and Cm,2, such that the pairs (CP, Cm,1) and (CP, Cm,2) are diffeomorphic, but Cm,1 and Cm,2 have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of ”Dif=Def” and ”Dif=Iso” problems for plane irreducible cuspidal curves. In our examples, Cm,1 and Cm,2 ...