Component group of thep-new subvariety ofJ 0(M p )

For an abelian varietyA over ℚ _p , the special fibre in the Néron model ofA over ℤ _p is the extension of a finite group scheme over ℤ _p , called the group of connected components, by the connected component of identity. WhenA is the Jacobian variety of an algebraic curve, its component group has been calculated in many cases. We determine in this paper the component group of thep-new subvariety ofJ ₀(M _p ), forM>1 a positive integer andp≥5 a prime not dividingM. Such a subvariety is not the Jacobian of any obvious curve, but it is not clear if it can ever be realised as the Jacobian of a curve.     

On Eisenstein ideals and the cuspidal group of J0(N)

Let C_N be the cuspidal subgroup of the Jacobian J₀(N) for a square-free integer N > 6. For any Eisenstein maximal ideal m of the Hecke ring of level N, we show that C_N[m] ≠ 0. To prove this, we calculate the index of an Eisenstein ideal I contained in m by computing the order of the cuspidal divisor annihilated by I.     

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

For a positive integer N, let $X_0(N)$ be the modular curve over $\mathbf{Q}$ and $J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of $J_0(N)$ is equal to the rational cuspidal divisor class group of $X_0(N)$ when $N=p^{2}M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m,h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.     

2-dimensional simple factors ofJ 0(N)

In this paper we investigate abelian varietyA _f which is derived from a newformf ∈ S ₂(Г₀(N)) an is ℚ-simple factors ofJac(X ₀ (N)). We will develop algorithms for computing the period matrix ofA _f and for determing whenA _f is principally polarized. IfA _f is 2-dimensional principally polarized, we give an algorithm for computing the associated hyperelliptic curveC withJac(C)≊A _f.     

The old subvariety ofJ 0(pq) and the Eisenstein kernel in Jacobians

Whenp, q are distinct odd primes, and γ:J ₀(p)²×J ₀(q)²→J ₀(pq) is the natural map defined by the degeneracy maps, Ribet [10] determined the odd part of the kernel of γ. We study the 2-primary part of this kernel through its intersection with the Eisenstein kernelJ ₀(p)[I _p )²×J ₀(q)[I _q ]². We determine this intersection forp≢1 mod 16,q≢1 mod 16, and also produce new elements of ker γ wheneverp≡9 mod 16 orq≡9 mod 16. These sharpen Ribet's results in [10].     

On the rational cuspidal subgroup and the rational torsion points of

For two distinct prime numbers , , we compute the rational cuspidal subgroup of and determine the -primary part of the rational torsion subgroup of the old subvariety of for most primes . Some results of Berkovic on the nontriviality of the Mordell-Weil group of some Eisenstein factors of are also refined.

     

On the commutator subgroup of a nonconnected central group

In this paperG denotes a central topologicalT ₂-group—G/Z(G) compact, whereZ(G) is the center. There are some results concerning compactness of the commutator subgroupG; in general (G)^– is compact ([3]), but not necessarilyG ([7]). If in additionG is a Lie group or ifG is connected,G is compact ([6], [5]). The purpose of this paper is to show, that if the componentG ₀ of the identity is open,G must be compact, and to give an example of a compact group with (G/G ₀) compact, whileG is not compact.Dedicated to Prof. R. Inzinger on his 70th birthday     

On the length of subgroup chains in the symmetric group

We prove that for n≥2, the length of every subgroup chain in S_n is at most 2n-3. The proof rests on an upper bound for the order of primitive permutation groups, due to Praeger and Saxl. The result has applications to worst case complexity estimates for permutation group algorithms.     

On the complexity of the (r|p)-centroid problem in the plane

In the (r∣p)-centroid problem, two players, called leader and follower, open facilities to service clients. We assume that clients are identified with their location on the Euclidean plane, and facilities can be opened anywhere in the plane. The leader opens p facilities. Later on, the follower opens r facilities. Each client patronizes the closest facility. In case of ties, the leader’s facility is preferred. The goal is to find p facilities for the leader to maximize his market share. We show that this Stackelberg game is \(\varSigma_{2}^{P}\) -hard. Moreover, we strengthen the previous results for the discrete case and networks. We show that the game is \(\varSigma_{2}^{P}\) -hard even for planar graphs for which the weights of the edges are Euclidean distances between vertices.     

关于wF(p,r,q)算子类

作为wA(p,r)算子类的一个推广,该文介绍了一类更广泛的算子类即 wF(p,r,q)算子类,它包含A(p,r)类而含于F(p,r,q)类之中, 进而考虑了该类算子的特征, 包含关系, 正规性和幂性质等等.