36,331 research outputs found
Exponential Carmichael function
Consider exponential Carmichael function such that
is multiplicative and , where
is usual Carmichael function. We discuss the value of , where runs over certain subsets of , and provide
bounds on the error term, using analytic methods and especially estimates of
.Comment: 9 page
Structure computation and discrete logarithms in finite abelian p-groups
We present a generic algorithm for computing discrete logarithms in a finite
abelian p-group H, improving the Pohlig-Hellman algorithm and its
generalization to noncyclic groups by Teske. We then give a direct method to
compute a basis for H without using a relation matrix. The problem of computing
a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian
group G is addressed, yielding a Monte Carlo algorithm to compute the structure
of G using O(|G|^0.5) group operations. These results also improve generic
algorithms for extracting pth roots in G.Comment: 23 pages, minor edit
A local-global principle for rational isogenies of prime degree
Let K be a number field. We consider a local-global principle for elliptic
curves E/K that admit (or do not admit) a rational isogeny of prime degree n.
For suitable K (including K=Q), we prove that this principle holds when n = 1
mod 4, and for n < 7, but find a counterexample when n = 7 for an elliptic
curve with j-invariant 2268945/128. For K = Q we show that, up to isomorphism,
this is the only counterexample.Comment: 11 pages, minor edits, to appear in Journal de Th\'eorie des Nombres
de Bordeau
- …