In two previous papers [AGM02,AGM08] we computed cohomology groups H(Γ0(N);C) for a range of levels N , where Γ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N . In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main fo...

‎There are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{Z}_{2}$ or $mathbb{Z}_{15}$. Still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of Sylow $p$-subgroups for each prime $p$, etc. In this...

Our aim here is to give a fairly self-contained exposition of some basic facts about the Iwahori-Hecke algebra H of a split p-adic group, including Bernstein’s presentation and description of the center, Macdonald’s formula, the CasselmanShalika formula, and the Kato-Lusztig formula. There are no new results here, and the same is essentially true of the proofs. We have been strongly influenced ...

Consider an elliptic curve E over a number field K. Suppose that has supersingular reduction at some prime $$\mathfrak {p}$$ of K lying above the rational p. We completely classify valuations $$p^n$$ -torsion points by valuation coefficient $$p{\text {th}}$$ division polynomial. This classification corrects error in earlier work Lozano-Robledo. As application, we find minimum necessary ramifica...

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k >2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p+...

Many cryptographic protocols and cryptosystems have been proposed to make use of prime order subgroups of Z n where n is the product of two large distinct primes. In this paper we analyse a number of such schemes. While these schemes were proposed to utilise the diiculty of factoring large integers or that of nding a trapdoor in Z n (for instance, the order of an RSA group), our analyses show m...

Assume N is prime, so X0(N)Q has two cusps; these are labelled 0 and ∞. The two cusps are Q-rational points on X0(N)Q and the divisor (0) − (∞) on X0(N)Q generates a finite cyclic subgroup C in J0(N)(Q) called the cuspidal divisor group. Denote by C/Z the finite flat subgroup scheme of J generated by C ⊂ J0(N)(Q). Let C be the FN -valued points of CZ (“the specialization” of C in J × FN ). It i...

‎there are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{z}_{2}$ or $mathbb{z}_{15}$. still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of sylow $p$-subgroups for each prime $p$, etc. in this...