اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدEigenvalue of Fricke involution on newforms of level 4 and of trivial character
56
0
0.0
(
0
)
تأليف
Yichao Zhang
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this note, we consider the newforms of integral weight, level 4 and of trivial character, and prove that all of them are actually level 1 forms of some non-Dirichlet character. As a byproduct, we can prove that all of them are eigenfunctions of the Fricke involution with eigenvalue -1.
قيم البحث
اقرأ أيضاً
In the spirit of Lehmers unresolved speculation on the nonvanishing of Ramanujans tau-function, it is natural to ask whether a fixed integer is a value of $tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, w
hich applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3leq ellleq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $tau(n)$ with $n>1$, we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 13, pm 17, -19, pm 23, pm 37, pm 691},$$ and assuming GRH we show for primes $ell$ that $$tau(n) ot in left { pm ell : 41leq ellleq 97 {textrm{with}} left(frac{ell}{5}right)=-1right} cup left { -11, -29, -31, -41, -59, -61, -71, -79, -89right}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $ell$, we prove that $pm ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{pm}(ell,m)=O_{ell}(m)$ and even level coprime to $ell,$ where $M^{pm}(ell,m)$ is effectively computable.
Let $M_k^!(Gamma_0^+(3))$ be the space of weakly holomorphic modular forms of weight $k$ for the Fricke group of level $3$. We introduce a natural basis for $M_k^!(Gamma_0^+(3))$ and prove that for almost all basis elements, all of their zeros in a f
undamental domain lie on the circle centered at 0 with radius $frac{1}{sqrt{3}}$.
We compute the E-polynomials of a family of twisted character varieties by proving they have polynomial count, and applying a result of N. Katz on the counting functions. To compute the number of GF(q)-points of these varieties as a function of q,
we used a formula of Frobenius. Our calculations made use of the character tables of Gl(n,q) and Sl(n,q), previously computed by J. A. Green and G. Lehrer, and a result of Hanlon on the Mobius function of a subposet of set-partitions. The Euler Characteristics of these character varieties are calculated with these polynomial.
We show that Selbergs eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a co
mbination of methods from asymptotic perturbation theory and number theory.
In this paper, we consider the distribution of the continuous paths of Dirichlet character sums modulo prime $q$ on the complex plane. We also find a limiting distribution as $q rightarrow infty$ using Steinhaus random multiplicative functions, stati
ng properties of this random process. This is motivated by Kowalski and Sawins work on Kloosterman paths.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
