Theta function squared is a weight $1$ modular formeisenstein part of theta functionModular forms on the...

Why is there so much iron?

How do you justify more code being written by following clean code practices?

When should a starting writer get his own webpage?

Help with identifying unique aircraft over NE Pennsylvania

Writing in a Christian voice

What is the tangent at a sharp point on a curve?

Unable to get newly inserted Product's Id using After Plugin for Catalog Product save controller method

Animating wave motion in water

How do researchers send unsolicited emails asking for feedback on their works?

UK Tourist Visa- Enquiry

Should a narrator ever describe things based on a characters view instead of fact?

Do people actually use the word "kaputt" in conversation?

"Marked down as someone wanting to sell shares." What does that mean?

How to determine the greatest d orbital splitting?

Can other pieces capture a threatening piece and prevent a checkmate?

How to balance a monster modification (zombie)?

The English Debate

When did hardware antialiasing start being available?

Can "few" be used as a subject? If so, what is the rule?

Hot air balloons as primitive bombers

How can an organ that provides biological immortality be unable to regenerate?

PTIJ: Where did Achashverosh's years wander off to?

What are the consequences of changing the number of hours in a day?

What are the rules for concealing thieves' tools (or items in general)?



Theta function squared is a weight $1$ modular form


eisenstein part of theta functionModular forms on the theta groupOn the square coeffecients of a modular formModular forms on $Gamma_0(N)$ with character in SageReverse the twisting of modular formCusp condition of modular forms of half-integral weightWriting an integer as a sum of two squares: Why is this a modular form of weight 1?*WHY* are half-integral weight modular forms defined on congruence subgroups of level 4N?Hecke operator on half integral weight modular formGenerating function for sum of two squares is a modular form













2












$begingroup$


Let
$$vartheta(tau) = sum_{ninmathbb{Z}}e^{pi in^2tau}.$$
I know that $vartheta$ satisfies the transfromation properties
$$vartheta(tau + 2) = vartheta(tau), quad varthetaleft(-frac{1}{tau}right) = sqrt{frac{tau}{i}}vartheta(tau).$$
What I am interested in is the transformation properties of
$$f(tau) = vartheta(2tau)^2.$$
I have seen it mentioned in some notes by Kevin Buzzard and elsewhere that $f(tau)$ is a weight $1$ modular form on the congruence subgroup $Gamma_0(4)$ with Nebentypus $chi$, the Dirichlet character coming from the Legendre symbol mod $4$. In other words,
$$fleft(frac{atau + b}{ctau + d}right) = chi(d)(ctau + d)f(tau), quad begin{pmatrix} a & b \ c & d end{pmatrix} in Gamma_0(4).$$
I cannot find a reference for this fact nor does it seem obvious from the transformation properties of $vartheta(tau)$. I know that given a modular form $g(tau)$ of weight $k$ on $Gamma_0(N)$ for some $N$, the function $g(2tau)^2$ is modular of weight $2k$ on $Gamma_0(2N)$. But the problem is that the subgroup on which $vartheta(tau)$ is modular (of weight $1/2$) is not one of the congruence subgroups $Gamma_0(N)$. I would greatly appreciate any help tracking down a reference for this fact.






number-theory modular-forms theta-functions







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    Let
    $$vartheta(tau) = sum_{ninmathbb{Z}}e^{pi in^2tau}.$$
    I know that $vartheta$ satisfies the transfromation properties
    $$vartheta(tau + 2) = vartheta(tau), quad varthetaleft(-frac{1}{tau}right) = sqrt{frac{tau}{i}}vartheta(tau).$$
    What I am interested in is the transformation properties of
    $$f(tau) = vartheta(2tau)^2.$$
    I have seen it mentioned in some notes by Kevin Buzzard and elsewhere that $f(tau)$ is a weight $1$ modular form on the congruence subgroup $Gamma_0(4)$ with Nebentypus $chi$, the Dirichlet character coming from the Legendre symbol mod $4$. In other words,
    $$fleft(frac{atau + b}{ctau + d}right) = chi(d)(ctau + d)f(tau), quad begin{pmatrix} a & b \ c & d end{pmatrix} in Gamma_0(4).$$
    I cannot find a reference for this fact nor does it seem obvious from the transformation properties of $vartheta(tau)$. I know that given a modular form $g(tau)$ of weight $k$ on $Gamma_0(N)$ for some $N$, the function $g(2tau)^2$ is modular of weight $2k$ on $Gamma_0(2N)$. But the problem is that the subgroup on which $vartheta(tau)$ is modular (of weight $1/2$) is not one of the congruence subgroups $Gamma_0(N)$. I would greatly appreciate any help tracking down a reference for this fact.






    number-theory modular-forms theta-functions







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      Let
      $$vartheta(tau) = sum_{ninmathbb{Z}}e^{pi in^2tau}.$$
      I know that $vartheta$ satisfies the transfromation properties
      $$vartheta(tau + 2) = vartheta(tau), quad varthetaleft(-frac{1}{tau}right) = sqrt{frac{tau}{i}}vartheta(tau).$$
      What I am interested in is the transformation properties of
      $$f(tau) = vartheta(2tau)^2.$$
      I have seen it mentioned in some notes by Kevin Buzzard and elsewhere that $f(tau)$ is a weight $1$ modular form on the congruence subgroup $Gamma_0(4)$ with Nebentypus $chi$, the Dirichlet character coming from the Legendre symbol mod $4$. In other words,
      $$fleft(frac{atau + b}{ctau + d}right) = chi(d)(ctau + d)f(tau), quad begin{pmatrix} a & b \ c & d end{pmatrix} in Gamma_0(4).$$
      I cannot find a reference for this fact nor does it seem obvious from the transformation properties of $vartheta(tau)$. I know that given a modular form $g(tau)$ of weight $k$ on $Gamma_0(N)$ for some $N$, the function $g(2tau)^2$ is modular of weight $2k$ on $Gamma_0(2N)$. But the problem is that the subgroup on which $vartheta(tau)$ is modular (of weight $1/2$) is not one of the congruence subgroups $Gamma_0(N)$. I would greatly appreciate any help tracking down a reference for this fact.






      number-theory modular-forms theta-functions







      share|cite|improve this question









      $endgroup$




      Let
      $$vartheta(tau) = sum_{ninmathbb{Z}}e^{pi in^2tau}.$$
      I know that $vartheta$ satisfies the transfromation properties
      $$vartheta(tau + 2) = vartheta(tau), quad varthetaleft(-frac{1}{tau}right) = sqrt{frac{tau}{i}}vartheta(tau).$$
      What I am interested in is the transformation properties of
      $$f(tau) = vartheta(2tau)^2.$$
      I have seen it mentioned in some notes by Kevin Buzzard and elsewhere that $f(tau)$ is a weight $1$ modular form on the congruence subgroup $Gamma_0(4)$ with Nebentypus $chi$, the Dirichlet character coming from the Legendre symbol mod $4$. In other words,
      $$fleft(frac{atau + b}{ctau + d}right) = chi(d)(ctau + d)f(tau), quad begin{pmatrix} a & b \ c & d end{pmatrix} in Gamma_0(4).$$
      I cannot find a reference for this fact nor does it seem obvious from the transformation properties of $vartheta(tau)$. I know that given a modular form $g(tau)$ of weight $k$ on $Gamma_0(N)$ for some $N$, the function $g(2tau)^2$ is modular of weight $2k$ on $Gamma_0(2N)$. But the problem is that the subgroup on which $vartheta(tau)$ is modular (of weight $1/2$) is not one of the congruence subgroups $Gamma_0(N)$. I would greatly appreciate any help tracking down a reference for this fact.






      number-theory modular-forms theta-functions




      number-theory modular-forms theta-functions






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Mar 7 at 4:58









      Ethan AlwaiseEthan Alwaise

      6,446717




      6,446717






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          $g = vartheta^2(2tau)$ is invariant under $Gamma_1(4)$ is easily checked: since $Gamma_1(4)$ is generated by
          $$begin{pmatrix}1 & 1 \ 0 & 1 end{pmatrix}qquad begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}$$
          Obviously $g$ is weight-$1$ invariant under the first one. For second one,
          $$begin{aligned}gleft[begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}right]_1 (tau)&= (4tau +1)^{-1}vartheta^2(frac{2tau}{4tau+1}) \&= (4tau +1)^{-1} frac{i(4tau+1)}{2tau}vartheta^2(-frac{4tau+1}{2tau})
          \& = frac{i}{2tau}vartheta^2(frac{-1}{2tau}) = vartheta^2(2tau)
          end{aligned}$$



          To check $g$ has nebentypus $chi$, it suffices to show $$gleft[begin{pmatrix}-1 & 0 \ 4 & -1 end{pmatrix}right]_1(tau)=-g(tau)$$
          I shall leave this checking to you.






          share|cite|improve this answer











          $endgroup$













            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%2f3138482%2ftheta-function-squared-is-a-weight-1-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









            1












            $begingroup$

            $g = vartheta^2(2tau)$ is invariant under $Gamma_1(4)$ is easily checked: since $Gamma_1(4)$ is generated by
            $$begin{pmatrix}1 & 1 \ 0 & 1 end{pmatrix}qquad begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}$$
            Obviously $g$ is weight-$1$ invariant under the first one. For second one,
            $$begin{aligned}gleft[begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}right]_1 (tau)&= (4tau +1)^{-1}vartheta^2(frac{2tau}{4tau+1}) \&= (4tau +1)^{-1} frac{i(4tau+1)}{2tau}vartheta^2(-frac{4tau+1}{2tau})
            \& = frac{i}{2tau}vartheta^2(frac{-1}{2tau}) = vartheta^2(2tau)
            end{aligned}$$



            To check $g$ has nebentypus $chi$, it suffices to show $$gleft[begin{pmatrix}-1 & 0 \ 4 & -1 end{pmatrix}right]_1(tau)=-g(tau)$$
            I shall leave this checking to you.






            share|cite|improve this answer











            $endgroup$


















              1












              $begingroup$

              $g = vartheta^2(2tau)$ is invariant under $Gamma_1(4)$ is easily checked: since $Gamma_1(4)$ is generated by
              $$begin{pmatrix}1 & 1 \ 0 & 1 end{pmatrix}qquad begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}$$
              Obviously $g$ is weight-$1$ invariant under the first one. For second one,
              $$begin{aligned}gleft[begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}right]_1 (tau)&= (4tau +1)^{-1}vartheta^2(frac{2tau}{4tau+1}) \&= (4tau +1)^{-1} frac{i(4tau+1)}{2tau}vartheta^2(-frac{4tau+1}{2tau})
              \& = frac{i}{2tau}vartheta^2(frac{-1}{2tau}) = vartheta^2(2tau)
              end{aligned}$$



              To check $g$ has nebentypus $chi$, it suffices to show $$gleft[begin{pmatrix}-1 & 0 \ 4 & -1 end{pmatrix}right]_1(tau)=-g(tau)$$
              I shall leave this checking to you.






              share|cite|improve this answer











              $endgroup$
















                1












                1








                1





                $begingroup$

                $g = vartheta^2(2tau)$ is invariant under $Gamma_1(4)$ is easily checked: since $Gamma_1(4)$ is generated by
                $$begin{pmatrix}1 & 1 \ 0 & 1 end{pmatrix}qquad begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}$$
                Obviously $g$ is weight-$1$ invariant under the first one. For second one,
                $$begin{aligned}gleft[begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}right]_1 (tau)&= (4tau +1)^{-1}vartheta^2(frac{2tau}{4tau+1}) \&= (4tau +1)^{-1} frac{i(4tau+1)}{2tau}vartheta^2(-frac{4tau+1}{2tau})
                \& = frac{i}{2tau}vartheta^2(frac{-1}{2tau}) = vartheta^2(2tau)
                end{aligned}$$



                To check $g$ has nebentypus $chi$, it suffices to show $$gleft[begin{pmatrix}-1 & 0 \ 4 & -1 end{pmatrix}right]_1(tau)=-g(tau)$$
                I shall leave this checking to you.






                share|cite|improve this answer











                $endgroup$



                $g = vartheta^2(2tau)$ is invariant under $Gamma_1(4)$ is easily checked: since $Gamma_1(4)$ is generated by
                $$begin{pmatrix}1 & 1 \ 0 & 1 end{pmatrix}qquad begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}$$
                Obviously $g$ is weight-$1$ invariant under the first one. For second one,
                $$begin{aligned}gleft[begin{pmatrix}1 & 0 \ 4 & 1 end{pmatrix}right]_1 (tau)&= (4tau +1)^{-1}vartheta^2(frac{2tau}{4tau+1}) \&= (4tau +1)^{-1} frac{i(4tau+1)}{2tau}vartheta^2(-frac{4tau+1}{2tau})
                \& = frac{i}{2tau}vartheta^2(frac{-1}{2tau}) = vartheta^2(2tau)
                end{aligned}$$



                To check $g$ has nebentypus $chi$, it suffices to show $$gleft[begin{pmatrix}-1 & 0 \ 4 & -1 end{pmatrix}right]_1(tau)=-g(tau)$$
                I shall leave this checking to you.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Mar 12 at 5:28

























                answered Mar 12 at 5:19









                piscopisco

                11.9k21743




                11.9k21743






























                    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.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3138482%2ftheta-function-squared-is-a-weight-1-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

                    Nidaros erkebispedøme

                    Image
                    For det lutherske bispedømet, sjå Nidaros bispedøme. Nidaros erkebispedøme var eit katolsk erkebispedøme med sete i Nidaros. Det vart oppretta av kong Olav Kyrre mellom 1070 og 1080 som bispedøme. Bispedømet vart opphøgd til erkebispedøme i 1153. Erkebispedømet vart innstifta av pave Anastasius IV i 1154, og var i funksjon fram til reformasjonen i 1537. Erkebiskopen sitt hovudsete var Erkebispegarden, rett ved domkyrkja Nidarosdomen. Frå 1152 til 1154 var Nicholas Breakspear i Skandinavia som paveleg utsending, og organiserte opprettinga av erkesetet. Innhaldsliste 1 Utbreiing og jurisdiksjon 2 Overhyrdar 2.1 Lydbiskopar 3 Biskopar 3.1 Biskopar 3.2 Erkebiskopar Utbreiing og jurisdiksjon | Kart over den nordiske kyrkjeprovinsen (1153-1387) Erkebispedømet omfatta i tillegg til Noreg øyane i vest i Noregsveldet. Nidaros var eit metropolitanerkebispedøme med fylgjande underliggjande bispedøme suffraganar: Bj...
                    Read more

                    Birsay

                    Image
                    Tettstaden Birsay på Orknøyane. Ruinane av jarleplasset om lag midt i biletet. (Fotografi frå 2012) Utsikt mot Birsay og Brough of Brisay (fotografi frå 2009) Birsay er ein tettstad i eit sokn (parish) med same namn, nordvest på øya Mainland [Note 1] på Orknøyane, om lag 31 km frå Kirkwall. Staden er omtala fleire stader i Orknøyingasoga [1] , der han er kalla «Byrgesherad» (norrønt: Birgisherað ). Namnet skriv seg truleg frå elvenamnet Birgisá [2] . Innhaldsliste 1 Den gamle hovudstaden 2 Bispesetet 3 Jarlepalasset 4 Kjelder og utfyllande opplysningar 4.1 Notar 4.2 Fotnotar 4.3 Bakgrunnsstoff Den gamle hovudstaden | Hovudartikkelen for dette emnet er Torfinn Sigurdsson . Soknet er i dag i hovudsak ei jordbrukskommune men er mest kjend for dei mange kulturminna i området. Tettstaden Birsay blir gjerne omtala som «den gamle hovudstaden på Orknøyane» [3] , og ein viser d...
                    Read more

                    Was Woodrow Wilson really a Liberal?Was World War I a war of liberals against authoritarians?Founding Fathers...

                    Image
                    Do VLANs within a subnet need to have their own subnet for router on a stick? What are the differences between the usage of 'it' and 'they'? Dragon forelimb placement How do we improve the relationship with a client software team that performs poorly and is becoming less collaborative? What typically incentivizes a professor to change jobs to a lower ranking university? How to write a macro that is braces sensitive? Modeling an IPv4 Address Writing rule stating superpower from different root cause is bad writing Can I ask the recruiters in my resume to put the reason why I am rejected? Approximately how much travel t...
                    Read more