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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

Citations:

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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