# Possible slope for a modular form

Let $f\colon \mathbb{H}_g \to \mathbb{C}$ be a Siegel modular form of weight $k$ with respect to $\Gamma_g$. Then, $f$ admits a Fourier expansion $f(Z) = \sum_T a(T) \exp(i\pi \mathop{tr} TZ)$, where $T$ varies amongst all even symmetrical semipositive definite matrices. The number $v := \frac{1}{2}\min_{x \in \mathbb{Z}^g \setminus 0} \{x^tTx \mid a(T) \neq 0\}$ is the vanishing order of $f$. A cusp form is a modular form with $v > 0$. (Check, for example, Slope of Integral Lattices, C.Poor, D.Yuen)

We define the slope of a Modular cusp form $\mathop{sl}(f) = \frac{k}{v}$, which is a rational positive number.

Is there a modular cusp form with slope $q$ for every rational positive $q$?

If $g$ is fixed, I think the answer is false (this is related to the minimale slope of $\mathcal{A}_g$, which is greater than 0 -and it is only 0 asymptotically). So a better questions should be the following:

If $g$ is fixed, is there a modular cusp form with slope $q$ for every rational number $q > \bar{q}$, for some $\bar{q} = \bar{q}(g) \in \mathbb{Q}$?

If the answer is known to be negative, how can I possibly prove that there is no Modular cusp form with slope $q$ for a $q$ which contradicts the claim? If the answer is known to be positive, how can I produce a modular form with slope $q$?

The idea behind this is to use modular forms to define slope of loci inside $\mathcal{A}_g$. For example, we have that the slope of the hyperelliptic locus is $8 + \frac{4}{g}$ since any modular form with slope $< 8 + 4/g$ vanishes on the hyperelliptic locus and this is sharp (we have a modular form with slope $8 + 4/g$ which does not vanish on the hyperelliptic locus -see R. Salvati-Manni "Slope of cusp forms and Theta series" JNT (2000)). The problem is that in general, modular forms could not be sharp enough to describe the slope of loci inside $\mathcal{A}_g$, i.e. we know that any modular form with slope $< q_1$ vanishes on a certain locus while any modular form with slope $> q_2$ does not vanish on the same locus, and there is no modular form with slope $q_1 < q < q_2$. I expect the modular forms to be sloppy (sorry for the pun!) for this purpose, but I think the main question is of some interest anyway.

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