## Note on Igusa’s cusp form of weight 35.(English)Zbl 1368.11043

The authors prove an interesting congruence for coefficients of (the constant $$4i$$ times) Igusa’s weight $$35$$ degree $$2$$ Siegel cusp form. This congruence is equivalent to the coefficients of its image under the theta operator (the Siegel analogue of the usual differential operator $$q\frac{d}{dq}$$) being divisible by $$23$$. The interested reader should also compare this result to a similar congruence in Satz 5 (a) of [S. Böcherer and S. Nagaoka, Math. Ann. 338, No. 2, 421–433 (2007; Zbl 1171.11029)].
The paper is organized as follows. In Section 2, the reader will find the basic definitions from the theory of Siegel modular forms, as well as a discussion of Igusa’s generators of the graded algebra of degree $$2$$ Siegel modular forms, which highlights the interest in the form studied here. They then recall a recently proved Sturm-type theorem for proving congruences of even weight degree $$2$$ Siegel modular forms by checking them for the first coefficients. They further introduce some new language for a convenient rephrasing of this result, and prove an analogue for odd weight Siegel modular forms. The other key input needed for the proof is a result of Böcherer and Nagaoka which states that the theta operators preserves the space of mod $$p$$ Siegel modular forms of degree $$2$$ (with the weight given explicitly). This is an analogue of a well-known result for the ordinary $$q$$-derivative in the elliptic case, which is an important result in Serre’s theory of modular forms mod $$p$$. These two results reduce the proof to a finite computation of the first few Fourier coefficients. This numerical calculation is carried out in Section 3, completing the proof.

### MSC:

 11F33 Congruences for modular and $$p$$-adic modular forms 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

### Keywords:

Siegel modular forms; Igusa cusp form; congruences

Zbl 1171.11029
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### References:

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