اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn mixed modular equations of degree 21
124
0
0.0
(
0
)
تأليف
S. Chandankumar
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In the proposed work, we establish a total of six new $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 7 and 21.These equations can be regarded as modular identities in the alternate theory of signature 3. As a consequence, several values of quotients of theta--function are evaluated.
قيم البحث
اقرأ أيضاً
In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $varphi(q)$ and $psi(q)$. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving $f(-q)$. As a c
onsequence, values of certain quotients of theta-function are evaluated.
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involvi
ng his theta--functions $varphi(q)$ and $psi(-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta--function and a continued fraction of order 12 are explicitly evaluated.
We study the relation between Hecke groups and the modular equations in Ramanujans theories of signature 2, 3 and 4. The solution $(alpha,beta)$ to the generalized modular equation satisfies a polynomial equation $P(alpha,beta)=0$ and we determine th
e degree in each of $alpha$ and $beta$ of the polynomial $P(alpha,beta)$ explicitly. We establish some mutually equivalent statements related to Hecke subgroups and modular equations, and prove that $(1-beta, 1-alpha)$ is also a solution to the generalized modular equation and $P(1-beta, 1-alpha)=0$.
In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form [y+Q_2(z)y+Q_3(z)y=0,quad zinmathbb{H}={zinmathbb{C} ,|,operatorname{Im}z>0 },] where $Q_2(z)$ and $Q_3(z)-frac12 Q_2(z)$ are meromorph
ic modular forms on $mathrm{SL}(2,mathbb{Z})$ of weight $4$ and $6$, respectively. We show that any quasimodular form of depth $2$ on $mathrm{SL}(2,mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $hat{rho}: mathrm{SL}(2,mathbb{Z})tomathrm{SL}(3,mathbb{C})$ for this MODE and give the necessary and sufficient condition for the irreducibility (resp. reducibility) of the representation. We show that the irreducibility yields the quasimodularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain $mathrm{SU}(3)$ Toda systems. Note that the $mathrm{SU}(N+1)$ Toda systems are the classical Plucker infinitesimal formulas for holomorphic maps from a Riemann surface to $mathbb{CP}^N$.
In this paper, we consider the first negative eigenvalue of eigenforms of half-integral weight k + 1/2 and obtain an almost type bound.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
