Definition of the weight $k$ hyperbolic Laplacian












1












$begingroup$


I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $mathbb C$-valued functions on the upper half plane (Chapter 2.1):



enter image description here



I was not able to verify (1.4) by the way. Maybe I made a mistake, but from what I calculated, the first equality was not correct. I was wondering whether one of the definitions (1.1), (1.2), (1.3) was not written correctly.



Edit: I have checked again, and (1.4) is actually correct.



Wikipedia defines the weight $k$ non-Euclidean Laplacian by



$$Delta_k = -y^2( frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2}) + iky (frac{partial}{partial x} + i frac{partial}{partial y})$$



Which definition is correct?






number-theory reference-request modular-forms laplacian differential-operators







share|cite|improve this question











$endgroup$












  • $begingroup$
    Just as a comment, there are some errata on that page (129): sporadic.stanford.edu/bump/match/errata.html
    $endgroup$
    – Zachary Selk
    Dec 22 '18 at 4:33










  • $begingroup$
    Those errors seem to have been fixed in my edition
    $endgroup$
    – D_S
    Dec 22 '18 at 4:39










  • $begingroup$
    Won't the first one lead to if $f(z)$ is smooth and $1$-periodic then $f(z) = sum_n a_n(y) e^{2i pi n x}$ and if it is in the kernel of $Delta_k$ then $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ where $partial_y^2 h_k(y) = h_k(y)+ h_k(y) C_k/ y$
    $endgroup$
    – reuns
    Dec 22 '18 at 5:51












  • $begingroup$
    I don't know..what is $(2pi|n|y)$?
    $endgroup$
    – D_S
    Dec 22 '18 at 17:19










  • $begingroup$
    $h_k(t)$ evaluated at $t = 2 pi N y$ with $N = |n|$. And $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ is what we need for the Mellin transform of $f(x+i.)-c_0$ being a Dirichlet series
    $endgroup$
    – reuns
    Dec 22 '18 at 23:38


















1












$begingroup$


I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $mathbb C$-valued functions on the upper half plane (Chapter 2.1):



enter image description here



I was not able to verify (1.4) by the way. Maybe I made a mistake, but from what I calculated, the first equality was not correct. I was wondering whether one of the definitions (1.1), (1.2), (1.3) was not written correctly.



Edit: I have checked again, and (1.4) is actually correct.



Wikipedia defines the weight $k$ non-Euclidean Laplacian by



$$Delta_k = -y^2( frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2}) + iky (frac{partial}{partial x} + i frac{partial}{partial y})$$



Which definition is correct?






number-theory reference-request modular-forms laplacian differential-operators







share|cite|improve this question











$endgroup$












  • $begingroup$
    Just as a comment, there are some errata on that page (129): sporadic.stanford.edu/bump/match/errata.html
    $endgroup$
    – Zachary Selk
    Dec 22 '18 at 4:33










  • $begingroup$
    Those errors seem to have been fixed in my edition
    $endgroup$
    – D_S
    Dec 22 '18 at 4:39










  • $begingroup$
    Won't the first one lead to if $f(z)$ is smooth and $1$-periodic then $f(z) = sum_n a_n(y) e^{2i pi n x}$ and if it is in the kernel of $Delta_k$ then $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ where $partial_y^2 h_k(y) = h_k(y)+ h_k(y) C_k/ y$
    $endgroup$
    – reuns
    Dec 22 '18 at 5:51












  • $begingroup$
    I don't know..what is $(2pi|n|y)$?
    $endgroup$
    – D_S
    Dec 22 '18 at 17:19










  • $begingroup$
    $h_k(t)$ evaluated at $t = 2 pi N y$ with $N = |n|$. And $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ is what we need for the Mellin transform of $f(x+i.)-c_0$ being a Dirichlet series
    $endgroup$
    – reuns
    Dec 22 '18 at 23:38
















1












1








1





$begingroup$


I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $mathbb C$-valued functions on the upper half plane (Chapter 2.1):



enter image description here



I was not able to verify (1.4) by the way. Maybe I made a mistake, but from what I calculated, the first equality was not correct. I was wondering whether one of the definitions (1.1), (1.2), (1.3) was not written correctly.



Edit: I have checked again, and (1.4) is actually correct.



Wikipedia defines the weight $k$ non-Euclidean Laplacian by



$$Delta_k = -y^2( frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2}) + iky (frac{partial}{partial x} + i frac{partial}{partial y})$$



Which definition is correct?






number-theory reference-request modular-forms laplacian differential-operators







share|cite|improve this question











$endgroup$




I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $mathbb C$-valued functions on the upper half plane (Chapter 2.1):



enter image description here



I was not able to verify (1.4) by the way. Maybe I made a mistake, but from what I calculated, the first equality was not correct. I was wondering whether one of the definitions (1.1), (1.2), (1.3) was not written correctly.



Edit: I have checked again, and (1.4) is actually correct.



Wikipedia defines the weight $k$ non-Euclidean Laplacian by



$$Delta_k = -y^2( frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2}) + iky (frac{partial}{partial x} + i frac{partial}{partial y})$$



Which definition is correct?






number-theory reference-request modular-forms laplacian differential-operators




number-theory reference-request modular-forms laplacian differential-operators






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 26 '18 at 5:40







D_S

















asked Dec 22 '18 at 4:04









D_SD_S

13.4k51551




13.4k51551












  • $begingroup$
    Just as a comment, there are some errata on that page (129): sporadic.stanford.edu/bump/match/errata.html
    $endgroup$
    – Zachary Selk
    Dec 22 '18 at 4:33










  • $begingroup$
    Those errors seem to have been fixed in my edition
    $endgroup$
    – D_S
    Dec 22 '18 at 4:39










  • $begingroup$
    Won't the first one lead to if $f(z)$ is smooth and $1$-periodic then $f(z) = sum_n a_n(y) e^{2i pi n x}$ and if it is in the kernel of $Delta_k$ then $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ where $partial_y^2 h_k(y) = h_k(y)+ h_k(y) C_k/ y$
    $endgroup$
    – reuns
    Dec 22 '18 at 5:51












  • $begingroup$
    I don't know..what is $(2pi|n|y)$?
    $endgroup$
    – D_S
    Dec 22 '18 at 17:19










  • $begingroup$
    $h_k(t)$ evaluated at $t = 2 pi N y$ with $N = |n|$. And $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ is what we need for the Mellin transform of $f(x+i.)-c_0$ being a Dirichlet series
    $endgroup$
    – reuns
    Dec 22 '18 at 23:38




















  • $begingroup$
    Just as a comment, there are some errata on that page (129): sporadic.stanford.edu/bump/match/errata.html
    $endgroup$
    – Zachary Selk
    Dec 22 '18 at 4:33










  • $begingroup$
    Those errors seem to have been fixed in my edition
    $endgroup$
    – D_S
    Dec 22 '18 at 4:39










  • $begingroup$
    Won't the first one lead to if $f(z)$ is smooth and $1$-periodic then $f(z) = sum_n a_n(y) e^{2i pi n x}$ and if it is in the kernel of $Delta_k$ then $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ where $partial_y^2 h_k(y) = h_k(y)+ h_k(y) C_k/ y$
    $endgroup$
    – reuns
    Dec 22 '18 at 5:51












  • $begingroup$
    I don't know..what is $(2pi|n|y)$?
    $endgroup$
    – D_S
    Dec 22 '18 at 17:19










  • $begingroup$
    $h_k(t)$ evaluated at $t = 2 pi N y$ with $N = |n|$. And $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ is what we need for the Mellin transform of $f(x+i.)-c_0$ being a Dirichlet series
    $endgroup$
    – reuns
    Dec 22 '18 at 23:38


















$begingroup$
Just as a comment, there are some errata on that page (129): sporadic.stanford.edu/bump/match/errata.html
$endgroup$
– Zachary Selk
Dec 22 '18 at 4:33




$begingroup$
Just as a comment, there are some errata on that page (129): sporadic.stanford.edu/bump/match/errata.html
$endgroup$
– Zachary Selk
Dec 22 '18 at 4:33












$begingroup$
Those errors seem to have been fixed in my edition
$endgroup$
– D_S
Dec 22 '18 at 4:39




$begingroup$
Those errors seem to have been fixed in my edition
$endgroup$
– D_S
Dec 22 '18 at 4:39












$begingroup$
Won't the first one lead to if $f(z)$ is smooth and $1$-periodic then $f(z) = sum_n a_n(y) e^{2i pi n x}$ and if it is in the kernel of $Delta_k$ then $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ where $partial_y^2 h_k(y) = h_k(y)+ h_k(y) C_k/ y$
$endgroup$
– reuns
Dec 22 '18 at 5:51






$begingroup$
Won't the first one lead to if $f(z)$ is smooth and $1$-periodic then $f(z) = sum_n a_n(y) e^{2i pi n x}$ and if it is in the kernel of $Delta_k$ then $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ where $partial_y^2 h_k(y) = h_k(y)+ h_k(y) C_k/ y$
$endgroup$
– reuns
Dec 22 '18 at 5:51














$begingroup$
I don't know..what is $(2pi|n|y)$?
$endgroup$
– D_S
Dec 22 '18 at 17:19




$begingroup$
I don't know..what is $(2pi|n|y)$?
$endgroup$
– D_S
Dec 22 '18 at 17:19












$begingroup$
$h_k(t)$ evaluated at $t = 2 pi N y$ with $N = |n|$. And $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ is what we need for the Mellin transform of $f(x+i.)-c_0$ being a Dirichlet series
$endgroup$
– reuns
Dec 22 '18 at 23:38






$begingroup$
$h_k(t)$ evaluated at $t = 2 pi N y$ with $N = |n|$. And $f(z) = c_0+sum_{nne 0} c_n h_k(2pi |n|y) e^{2i pi n x}$ is what we need for the Mellin transform of $f(x+i.)-c_0$ being a Dirichlet series
$endgroup$
– reuns
Dec 22 '18 at 23:38












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