Reformulating Theta Function Symmetry as a Modular Form
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If $theta$ is the Jacobi theta function $theta(tau) = sum e^{pi i n^2 tau}$, then $theta$ satisfies the Modular symmetries $theta(tau + 2) = theta(tau)$ and $theta(-1/tau) = sqrt{-i tau} cdot theta(tau)$. Even if we square things, this isn't really completely the symmetry that a modular form should satisfy, i.e. $theta^2(-1/z) = - i tau cdot theta(tau)$ whereas a modular form $f$ of weight one should satisfy $f(-1/tau) = tau f(tau)$. Is there a standard way of working with the $theta$ function so we can treat it, or powers of it, as actual modular forms?
analytic-number-theory modular-forms
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
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add a comment |
$begingroup$
If $theta$ is the Jacobi theta function $theta(tau) = sum e^{pi i n^2 tau}$, then $theta$ satisfies the Modular symmetries $theta(tau + 2) = theta(tau)$ and $theta(-1/tau) = sqrt{-i tau} cdot theta(tau)$. Even if we square things, this isn't really completely the symmetry that a modular form should satisfy, i.e. $theta^2(-1/z) = - i tau cdot theta(tau)$ whereas a modular form $f$ of weight one should satisfy $f(-1/tau) = tau f(tau)$. Is there a standard way of working with the $theta$ function so we can treat it, or powers of it, as actual modular forms?
analytic-number-theory modular-forms
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
$endgroup$
$begingroup$
The most complete way is in Shimura's papers. Now the first theorem in the theory of modular forms is that $theta(2tau)^4$ is a weight $2$ modular form for $Gamma_0(4)$ thus a sum of two Eisenstein series (with known multiplicative coefficients), see first chapter of Diamond&shurman's book.
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– reuns
Dec 2 '18 at 23:57
add a comment |
$begingroup$
If $theta$ is the Jacobi theta function $theta(tau) = sum e^{pi i n^2 tau}$, then $theta$ satisfies the Modular symmetries $theta(tau + 2) = theta(tau)$ and $theta(-1/tau) = sqrt{-i tau} cdot theta(tau)$. Even if we square things, this isn't really completely the symmetry that a modular form should satisfy, i.e. $theta^2(-1/z) = - i tau cdot theta(tau)$ whereas a modular form $f$ of weight one should satisfy $f(-1/tau) = tau f(tau)$. Is there a standard way of working with the $theta$ function so we can treat it, or powers of it, as actual modular forms?
analytic-number-theory modular-forms
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
$endgroup$
If $theta$ is the Jacobi theta function $theta(tau) = sum e^{pi i n^2 tau}$, then $theta$ satisfies the Modular symmetries $theta(tau + 2) = theta(tau)$ and $theta(-1/tau) = sqrt{-i tau} cdot theta(tau)$. Even if we square things, this isn't really completely the symmetry that a modular form should satisfy, i.e. $theta^2(-1/z) = - i tau cdot theta(tau)$ whereas a modular form $f$ of weight one should satisfy $f(-1/tau) = tau f(tau)$. Is there a standard way of working with the $theta$ function so we can treat it, or powers of it, as actual modular forms?
analytic-number-theory modular-forms
analytic-number-theory modular-forms
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
asked Dec 2 '18 at 23:39
Jacob DensonJacob Denson
772313
772313
$begingroup$
The most complete way is in Shimura's papers. Now the first theorem in the theory of modular forms is that $theta(2tau)^4$ is a weight $2$ modular form for $Gamma_0(4)$ thus a sum of two Eisenstein series (with known multiplicative coefficients), see first chapter of Diamond&shurman's book.
$endgroup$
– reuns
Dec 2 '18 at 23:57
add a comment |
$begingroup$
The most complete way is in Shimura's papers. Now the first theorem in the theory of modular forms is that $theta(2tau)^4$ is a weight $2$ modular form for $Gamma_0(4)$ thus a sum of two Eisenstein series (with known multiplicative coefficients), see first chapter of Diamond&shurman's book.
$endgroup$
– reuns
Dec 2 '18 at 23:57
$begingroup$
The most complete way is in Shimura's papers. Now the first theorem in the theory of modular forms is that $theta(2tau)^4$ is a weight $2$ modular form for $Gamma_0(4)$ thus a sum of two Eisenstein series (with known multiplicative coefficients), see first chapter of Diamond&shurman's book.
$endgroup$
– reuns
Dec 2 '18 at 23:57
$begingroup$
The most complete way is in Shimura's papers. Now the first theorem in the theory of modular forms is that $theta(2tau)^4$ is a weight $2$ modular form for $Gamma_0(4)$ thus a sum of two Eisenstein series (with known multiplicative coefficients), see first chapter of Diamond&shurman's book.
$endgroup$
– reuns
Dec 2 '18 at 23:57
add a comment |
1 Answer
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The function $theta^2(z)$ is a weight $1$ modular form on $Gamma_0(4)$ with character $chi_{-1}$. That is, it satisfies
$$ theta^2(gamma z) = left( tfrac{-1}{d} right) (cz + d) theta^2(z), qquad gamma = left(begin{smallmatrix}a&b\c&dend{smallmatrix}right).$$
This fits nicely in the general philosophy of modular forms with character or modular forms with nebentypus.
I should note that one can also study $theta(z)$ as a half-integral weight modular form on a double-cover of $Gamma_0(4)$.
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
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1 Answer
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1 Answer
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$begingroup$
The function $theta^2(z)$ is a weight $1$ modular form on $Gamma_0(4)$ with character $chi_{-1}$. That is, it satisfies
$$ theta^2(gamma z) = left( tfrac{-1}{d} right) (cz + d) theta^2(z), qquad gamma = left(begin{smallmatrix}a&b\c&dend{smallmatrix}right).$$
This fits nicely in the general philosophy of modular forms with character or modular forms with nebentypus.
I should note that one can also study $theta(z)$ as a half-integral weight modular form on a double-cover of $Gamma_0(4)$.
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
$endgroup$
add a comment |
$begingroup$
The function $theta^2(z)$ is a weight $1$ modular form on $Gamma_0(4)$ with character $chi_{-1}$. That is, it satisfies
$$ theta^2(gamma z) = left( tfrac{-1}{d} right) (cz + d) theta^2(z), qquad gamma = left(begin{smallmatrix}a&b\c&dend{smallmatrix}right).$$
This fits nicely in the general philosophy of modular forms with character or modular forms with nebentypus.
I should note that one can also study $theta(z)$ as a half-integral weight modular form on a double-cover of $Gamma_0(4)$.
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
$endgroup$
add a comment |
$begingroup$
The function $theta^2(z)$ is a weight $1$ modular form on $Gamma_0(4)$ with character $chi_{-1}$. That is, it satisfies
$$ theta^2(gamma z) = left( tfrac{-1}{d} right) (cz + d) theta^2(z), qquad gamma = left(begin{smallmatrix}a&b\c&dend{smallmatrix}right).$$
This fits nicely in the general philosophy of modular forms with character or modular forms with nebentypus.
I should note that one can also study $theta(z)$ as a half-integral weight modular form on a double-cover of $Gamma_0(4)$.
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
$endgroup$
The function $theta^2(z)$ is a weight $1$ modular form on $Gamma_0(4)$ with character $chi_{-1}$. That is, it satisfies
$$ theta^2(gamma z) = left( tfrac{-1}{d} right) (cz + d) theta^2(z), qquad gamma = left(begin{smallmatrix}a&b\c&dend{smallmatrix}right).$$
This fits nicely in the general philosophy of modular forms with character or modular forms with nebentypus.
I should note that one can also study $theta(z)$ as a half-integral weight modular form on a double-cover of $Gamma_0(4)$.
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
answered Dec 2 '18 at 23:59
davidlowryduda♦davidlowryduda
74.4k7118252
74.4k7118252
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$begingroup$
The most complete way is in Shimura's papers. Now the first theorem in the theory of modular forms is that $theta(2tau)^4$ is a weight $2$ modular form for $Gamma_0(4)$ thus a sum of two Eisenstein series (with known multiplicative coefficients), see first chapter of Diamond&shurman's book.
$endgroup$
– reuns
Dec 2 '18 at 23:57