## nt.number theory – Ekedahl sieve for composite moduli

The following version of a theorem of Ekedahl, known as the Ekedahl sieve, can be found in this paper by Bhargava and Shankar for example, is stated as follows: let $$B$$ be a compact subset of $$mathbb{R}^n$$ having finite measure, and let $$Y$$ be any closed subscheme of $$mathbb{A}^n(mathbb{Z})$$ having co-dimension $$k geq 1$$. Let $$r, M$$ be positive real numbers. Then

$$displaystyle # {mathbf{a} in rB cap mathbb{Z}^n : mathbf{a} pmod{p} in Y(mathbb{F}_p) text{ for some prime } p > M }$$

$$= O left( frac{r^n}{M^{k-1} log M} + r^{n-k+1} right)$$

with the implied constant depending only on $$Y$$ and $$M$$.

If we replace the set on the left hand side with the set

$$displaystyle S_M(r) = {mathbf{a} in rB cap mathbb{Z}^n : mathbf{a} pmod{p} in Y(mathbb{F}_p) text{ for all } p | q text{ for some square-free } q > M }$$

can we obtain an upper bound of comparable strength to the original theorem?

Posted on Categories Articles

## nt.number theory – On the \$mathsf{LCM}\$ of a set of integers defined by moduli of powers

For integers $$a,b,t$$ define $$mathcal R_t(a,b)={qinmathbb Zcap(1,min(a^t,b^t)): a^tequiv b^tbmod q}$$ and $$mathsf{LCM}(mathcal R_t(a,b))$$ to be $$mathsf{LCM}$$ of all entries in $$mathcal R_t(a,b)$$.

Similar reasoning to On \$mathsf{LCM}\$ of a set of integers gives $$mathsf{LCM}(mathcal R_t(a,b))leqmathsf{LCM}(T_t(a,b))$$ where $$T_t(a,b)$$ is defined as $$T_t(a,b)=Big{qinmathbb Zcap(1,infty):q|Big((a-b)sum_{i=0}^{t-1}a^{t-1-i}b^iBig)Big}.$$

So $$mathsf{LCM}(mathcal R_t(a,b))=mathsf{LCM}(T_t(a,b))$$ holds.

If $$a,binbig(frac r2,rbig)$$ hold and are coprime then what is the probability $$mathsf{LCM}(mathcal R_t(a,b)) at some $$alphain(0,t)$$ and $$beta>0$$?

Posted on Categories Articles

## ag.algebraic geometry – moduli space of Nilepents Lie algebras

Repair a Nilpotent Lie algebra $$L$$ over a char 0 field $$k$$ that's of course graded, me. e. isomorphic to graduated algebra $$bar L$$ in connection with the lower central filtration.

I am interested in a reasonable description of $$M subset Hom (L otimes L, L)$$ consisting of algebras $$M$$ with an algebra isomorphism $$phi: bar M to L$$, Adequate description includes the action of $$GL (L)$$ and some constant compactification.

Perhaps someone knows much more and has already found a way to describe slices of nilpotent algebras for which the PBW morphism is, in a sense, a deformation of the Coalgebra map – as in the Lefevre-Hasegawa thesis; I think this remark needs some elaboration, which is best suited as a separate question. Therefore, references to any articles about this type of "Lie algebra discs with connection" are welcome.

Posted on Categories Articles