Zeros of modular forms.

  • #1
MathematicalPhysicist
Gold Member
4,328
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I am asked to find all the modular forms with weight k which don't have zeros on the upper half plane.

I know that a modular form with weight k is composed of an Eisenstein series with index k and a cusp form with weight k, and I have at my disposal the zeros formula for modular forms.

So I know that for k<12 (and k is obviously even cause Eisenstein series vanish for odd indices), they have zeros on the upper half plane, so I should be looking at modular form with weight 12m where m is positive integer.

Now if I have this equation for k=12m:
[tex]ord_{\infty} f + \sum_{p\neq i ,exp(2\pi i/3); p \in H/SL_2(Z)} ord_p f=m[/tex]
where H is the upper half plane, so the only f which it zero isn't in H is for [tex]ord_{\infty} f =m[/tex].

Is that enough?

Thanks.
 

Answers and Replies

  • #4
Petek
Gold Member
363
8
I looked at this, but don't have any useful information. You might get a better response at Math Stack Exchange.
 
  • #5
Stephen Tashi
Science Advisor
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1,416

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