## 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:

 [1] H. Aoki and T. Ibukiyama, Simple graded rings of Siegel modular forms, differential operators and Borcherds products , Int. J. Math. 16 (2005), 249-279. · Zbl 1068.11030 [2] S. Böcherer, Über gewisse Siegelsche Modulformen zweiten Grades , Math. Ann. 261 (1982), 23-41. · Zbl 0503.10017 [3] S. Böcherer and S. Nagaoka, On mod $$p$$ properties of Siegel modular forms , Math. Ann. 338 (2007), 421-433. · Zbl 1171.11029 [4] D. Choi, Y. Choie and T. Kikuta, Sturm type theorem for Siegel modular forms of genus $$2$$ modulo $$p$$ , Acta Arith. 158 (2013), 129-139. · Zbl 1288.11046 [5] J.-I. Igusa, On Siegel modular forms of genus two , Amer. J. Math. 84 (1962), 175-200; II, ibid. 86 (1964), 392-412. · Zbl 0133.33301 [6] —-, On the ring of modular forms of degree two over $$\boldsymbol{Z}$$ , Amer. J. Math. 101 (1979), 149-183. · Zbl 0415.14026 [7] C. Poor and D.S. Yuen, Paramodular cusp forms , arXiv: · Zbl 1392.11028 [8] J.-P. Serre, Formes modulaires et fonctions zêta $$p$$-adiques , Modular functions of one variable III, Lect. Notes Math. 350 (1972), 191-268.
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