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The definition of the Shimura lift of a classical cusp form is well documented. Zagier and Kohnen define a modified version of the lift for a cusp form $g(z)=\sum a(n)q^n \in S_{k+1/2}^{+}(4)$ in the Kohnen plus space for a fundamental discriminant $D$ with $(-1)^kD>0$ as

$$ S_D^+ (g(z)) = \sum_{n=1}^\infty \left( \sum_{d|n} \left(\frac{D}{d}\right)d^{k-1}a\left(\frac{n^2|D|}{d^2}\right)\right)q^n.$$

(Edit: Note that this definition is a slight modification of the original definition given by Shimura. It is defined only for forms in the plus space meeting the restriction $(-1)^kD>0$ and it includes a slightly different character than the original definition. The key consequence of this modified definition is that it lifts the form all the way down to level $N$, instead of just to level $2N$.)

They comment later in the same paper that the definition of the Shimura lift may be extended to modular forms in $M_{k+1/2}^+(4)$ by adding in a constant term of

$$\frac{1}{2}L_D(1-k)a(0)$$

to the above definition, where $L_D(1-k)$ is the L-series twisted by discriminant $D$ evaluated at $1-k$ by analytic continuation.

My question is the following: Is there a similar way to extend the definition of the Shimura lift to classical modular forms in $M_{k+1/2}^+(4N)$, in the plus space, for higher levels, with $N$ odd?

I have looked around somewhat in the literature; there are many sources which explain how to generalize the Shimura lift to cusp forms of higher level, but none seem to explain if/how you can extend the lift to non-cuspidal forms. If anyone knows a good source to read about this in the literature, I would appreciate being pointed in the right direction. If not, perhaps a summary of why the lift has not been extended this way/what the difficulties are.

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1 Answer 1

The approach due to Shintani and Niwa expresses Shimura's lift as an integral operator, where the kernel is given by a particular theta function.

A great resource for this theory is Barry Cipra's paper (and the references therein):

On the Niwa-Shintani theta kernel lifting of modular forms. Nagoya Math. J. (91) 1983

In particular, you can define a "Shimura lift" for any kind of modular form for which integrating against this theta kernel makes sense - Cipra discusses exactly this issue in Proposition 2.8.

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Thank you for bringing up this reference; it is very useful information. Note however, that the definition that Zagier and Kohnen use is actually a slightly modified version of that used by Shimura, Cipra and others. Cipra includes the character $\chi(n)$ from the original form, which implies that this is a level $4N$ character. Zagier and Kohnen do include a character, but this character is only a level $N$ character (in the paper I referenced N=1, but they later do it for arbitrary odd $N$). I will edit my question to more accurately reflect this. –  Nathan Green Jun 11 '13 at 21:47
    
Do you know if anyone has done any further research using this modified definition of the Shimura lift? –  Nathan Green Jun 11 '13 at 21:47

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