ﻻ يوجد ملخص باللغة العربية
We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory and group theoretic techniques.
We formulate some properties of a conjectural object $X_{fun}(r,n)$ parametrizing Anderson t-motives of dimension $n$ and rank $r$. Namely, we give formulas for $goth p$-Hecke correspondences of $X_{fun}(r,n)$ and its reductions at $goth p$ (where $g
Let $q:=e^{2 pi iz}$, where $z in mathbb{H}$. For an even integer $k$, let $f(z):=q^hprod_{m=1}^{infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and
Let $j(z)$ be the modular $j$-invariant function. Let $tau$ be an algebraic number in the complex upper half plane $mathbb{H}$. It was proved by Schneider and Siegel that if $tau$ is not a CM point, i.e., $[mathbb{Q}(tau):mathbb{Q}] eq2$, then $j(tau
We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $rho$ is determined by an equality of an $m$-power character $gmapsto Tr(rho(g^m))$ for some natural number $m$. U
Some new results concerning the equation $sigma(N)=aM, sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.