"Maass forms"의 두 판 사이의 차이

2013년 3월 17일 (일) 13:04 판

Hyperbolic distribution problems and half-integral weight Maass forms
Automorphic forms correspond to representations that occur in $L_2(G/\Gamma)$. In the case when $G$ is $SL_2$, holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of $G$, while Maass wave forms correspond to (spherical vectors of) continuous series representations of G.

\[S(a,b;p)=\sum_{1\leq x\leq p-1}{\exp(2i\pi (ax+b\bar{x})/p)},\quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } p\]

\[K(a,b;m)=\sum_{0\leq x\leq m-1,\ gcd(x,m)=1 } e^{2\pi i (ax+bx^*)/m}\]

Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory
Lectures on modular functions of one complex variable (Tata Institute of Fundamental Research. Lectures on mathematics and physics. Mathematics, 29)
- Hans Maass, (pdf)

@@ 14번째 줄: / 14번째 줄: @@
 * used to estimate the Fourier coefficients of modular forms
-*  definition for prime p
+* definition for prime p
-;<math>S(a,b;p)=\sum_{1\leq x\leq p-1}{\exp(2i\pi (ax+b\bar{x})/p)},\quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } p</math><br>
+:<math>S(a,b;p)=\sum_{1\leq x\leq p-1}{\exp(2i\pi (ax+b\bar{x})/p)},\quad\text{where}\quad x\bar{x}\equiv 1\text{ mod } p</math><br>
-*  generally defined as
+* generally defined as
 :<math>K(a,b;m)=\sum_{0\leq x\leq m-1,\ gcd(x,m)=1 } e^{2\pi i (ax+bx^*)/m}</math><br>
 * http://blogs.ethz.ch/kowalski/2010/02/26/the-fourth-moment-of-kloosterman-sums/