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







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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


















2












$begingroup$


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







share|cite|improve this question









$endgroup$












  • $begingroup$
    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.
    $endgroup$
    – reuns
    Dec 2 '18 at 23:57
















2












2








2





$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













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asked Dec 2 '18 at 23:39









Jacob DensonJacob Denson

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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




















  • $begingroup$
    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












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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 $Gamma_0(4)$.






share|cite|improve this 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 $Gamma_0(4)$.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      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 $Gamma_0(4)$.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        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 $Gamma_0(4)$.






        share|cite|improve this answer









        $endgroup$



        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 $Gamma_0(4)$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 2 '18 at 23:59









        davidlowryduda♦davidlowryduda

        74.4k7118252




        74.4k7118252






























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