اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe Eigencurve is Proper
295
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove that the Coleman-Mazur eigencurve is proper over the weight space for any prime p and tame level N.
قيم البحث
اقرأ أيضاً
We prove that the eigencurve associated to a definite quaternion algebra over $QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint u
nion of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the $U_p$-slopes of points on each fixed connected component are proportional to the $p$-adic valuations of the parameter on weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves towards the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.
Let $s(n):= sum_{dmid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $kge 2$, the equivalence [ text{$n$ is $k$th powerfree} Longleftrightarrow text{$s(n)$ is $k$th powerfree} ] holds almost
always (meaning, on a set of asymptotic density $1$). We prove this for $kge 4$.
Using the methods developed for the proof that the 2-universality criterion is unique, we partially characterize criteria for the n-universality of positive-definite integer-matrix quadratic forms. We then obtain the uniqueness of Ohs 8-universality
criterion as an application of our characterization results.
We consider the distribution in residue classes modulo primes $p$ of Eulers totient function $phi(n)$ and the sum-of-proper-divisors function $s(n):=sigma(n)-n$. We prove that the values $phi(n)$, for $nle x$, that are coprime to $p$ are asymptotical
ly uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 le p le (log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $ple (log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.
Let $T colon mathbb{N}to mathbb{N}$ denote the $3x+1$ function, where $T(n)=n/2$ if $n$ is even, $T(n)=(3n+1)/2$ if $n$ is odd. As an accelerated version of $T$, we define a emph{jump} at $n ge 1$ by $textrm{jp}(n) = T^{(ell)}(n)$, where $ell$ is the
number of digits of $n$ in base 2. We present computational and heuristic evidence leading to surprising conjectures. The boldest one states that for any $n ge 2^{150}$, at most four jumps starting from $n$ are needed to fall below $n$, a strong form of the Collatz conjecture.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
