The Unreasonable Slightness of E2 over Imaginary Quadratic Rings
نویسنده
چکیده
It is almost always the case that the elementary matrices generate the special linear group SLn over a ring of integers in a number field. The only exceptions to this rule occur for SL2 over rings of integers in imaginary quadratic fields. The surprise is compounded by the fact that, in these cases when elementary generation fails, it actually fails rather badly: the group E2 generated by the elementary 2-by-2 matrices turns out to be an infinite-index, non-normal subgroup of SL2. We give an elementary proof of this strong failure of elementary generation for SL2 over imaginary quadratic rings.
منابع مشابه
Lower bounds for decision problems in imaginary, norm-Euclidean quadratic integer rings
We prove lower bounds for the complexity of deciding several relations in imaginary, normEuclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer rin...
متن کاملWitt rings of quadratically presentable fields
This paper introduces an approach to the axiomatic theory of quadratic forms based on {tmem{presentable}} partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of {tmem{quadratically p...
متن کاملSome classes of strongly clean rings
A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.
متن کاملOn polarised class groups of orders in quartic CM-fields
We give an explicit characterisation of pairs of orders in a quartic CM-field that admit the same polarised ideal class group structure. This generalises a simpler result for imaginary quadratic fields. We give applications to computing endomorphism rings of abelian surfaces over finite fields, and extending a completeness result of Murabayashi and Umegaki [13] to a list of abelian surfaces ove...
متن کاملSecure Accumulators from Euclidean Rings without Trusted Setup
Cryptographic accumulators are well-known to be useful in many situations. However, the most efficient accumulator (the RSA accumulator) it is not secure against a certificate authority who has herself selected the RSA modulus n. We generalize previous work and define the root accumulator in modules over Euclidean rings. We prove that the root accumulator is secure under two different pairs of ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- The American Mathematical Monthly
دوره 118 شماره
صفحات -
تاریخ انتشار 2011