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?

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 $mathsf{LCM}(mathcal R_t(a,b))<beta r^{t-alpha}$ at some $alphain(0,t)$ and $beta>0$?

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.