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So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and spits out the sum of $q$ raised to the $N(v)$ power, where the sum is over all vectors $v$ in the lattice, $q$ is equal to $exp(2 \pi i z)$, and $N(v)$ is the norm squared. We can regard this as a formal sum for the sake of this question. Let's call it $T(z)$.

I know there all kinds of identities about this function. For example if the lattice is even and unimodular, then $T(z)$ is a modular form. I am curious about the related function $T(-z)$. Is there an easy way to relate it to $T(z)$?

What I'm really really interested in is the quotient $T(z)/T(-z)$. I know there are people who know this stuff way better then I do, can anyone help? Is it easier in the unimodular case?

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I'm having trouble understanding this, because under your definition, T(-z) requires T to be defined on the lower half-plane. The series doesn't converge there.

Also, N(v) should be half of the squared norm. Equivalently, N(v) should be the value of the quadratic form that defines the inner product on the lattice.

I can think of one interpretation, where the theta function is actually a Jacobi theta function, which is a Jacobi form, i.e., a section of a line bundle on the universal elliptic curve. This can be viewed as a function T(t,z) on HxC, with some invariance properties under translation by lattice elements in C and SL(2,Z) transformations in H. Then you can negate the z variable.

Eichler and Zagier have a book on Jacobi forms, called The theory of Jacobi forms.

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  • $\begingroup$ I see your point. Yes, I meant for N(v) to be the quadratic form that defines the inner product on the lattice; I forgot the factor of one-half. I think there was another mistake. I'll go fix this a bit. $\endgroup$
    – Chris Schommer-Pries
    Oct 14, 2009 at 16:36
  • $\begingroup$ Actually, I decided to leave it as it was. So does T(-z) then make sense on the lower half plane? or is the convergence issue worse then that. What if z is real? Can we make sense of T(z)/T(-z) then? $\endgroup$
    – Chris Schommer-Pries
    Oct 14, 2009 at 16:47
  • $\begingroup$ I don't know if it is possible to analytically continue T to the lower half plane. Modular forms tend to yield essential singularities at the boundary of the q-disc - pick a Mobius transformation that takes infinity to a rational number, and feed infinity to a positive weight transformation formula. I'm afraid I also don't know what happens if you try to feed in irrational real numbers. I've heard people mention modular forms definable on the lower half plane, maybe using GL2 instead of SL2, but I know basically nothing there. $\endgroup$
    – S. Carnahan ♦
    Oct 14, 2009 at 17:15

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