Reformulating Theta Function Symmetry as a Modular Form












2














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?










share|cite|improve this question






















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


















2














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?










share|cite|improve this question






















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
















2












2








2







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?










share|cite|improve this question













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













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 2 '18 at 23:39









Jacob Denson

772313




772313












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




















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


















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.
– reuns
Dec 2 '18 at 23:57






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.
– reuns
Dec 2 '18 at 23:57












1 Answer
1






active

oldest

votes


















2














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





















    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3023381%2freformulating-theta-function-symmetry-as-a-modular-form%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2














    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


























      2














      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
























        2












        2








        2






        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












        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

        74.3k7117251




        74.3k7117251






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3023381%2freformulating-theta-function-symmetry-as-a-modular-form%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Berounka

            Sphinx de Gizeh

            Different font size/position of beamer's navigation symbols template's content depending on regular/plain...