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Gaussian hypergeometric functions and traces of Hecke operators. (English) Zbl 1088.11029

The paper provides an impression of the miracles of modularity, relating objects which only recently seemed to be unrelated. Let \(p\) denote an odd prime. J. Greene [Trans. Am. Math. Soc. 301, 77–101 (1987; Zbl 0629.12017)] introduced Gaussian hypergeometric functions over the field \(\mathbb{F}_p\). It turned out that they yield formulas for Fourier coefficients of certain modular forms [M. Koike, Hiroshima Math. J. 22, No. 3, 461–467 (1992; Zbl 0784.11057); K. Ono, Trans. Am. Math. Soc. 350, No. 3, 1205–1223 (1998; Zbl 0910.11054)]. In the paper under review the authors exhibit trace formulas for the action of Hecke operators in terms of Gaussian hypergeometric functions over \(\mathbb{F}_p\). A special class of these functions is defined by \[ _{n+1} F_n (x)=\frac{p}{p-1}\cdot \sum_\chi{\varphi\chi\choose \chi}^{n+1}\cdot \chi(x) \] for \(x\in\mathbb{F}_p\), where \(\chi\) runs through all (multiplicative) characters of \(\mathbb{F}_p^\times\), \(\varphi (x)=(\frac xp)\) is the quadratic character modulo \(p\), and for any two characters \(\psi\) and \(\chi\) of \(\mathbb{F}^\times_p\), \[ {\psi\choose\chi}=\frac 1p \cdot\sum_{x\in\mathbb{F}_p} \psi(x)\overline\chi (x-1) \] is the normalized Jacobi sum. Starting from Hijikata’s version of the Eichler-Selberg trace formula, the authors express the trace of the Hecke operator \(T_p\) on the spaces \(S_k^{new} (\Gamma_0(8))\) of newforms of even weight \(k\) on the group \(\Gamma_0(8))\) in terms of values of \(_3F_2\). Their formula is recursive in \(k\) and can be written as \[ \varepsilon_k(p)+H_k(p)+ \sum^{k/2-1}_{j=0}p^j\cdot c_j(k/2-1)\cdot\text{Tr}^{\text{new}}_{k-2j}\bigl(\Gamma_0(8),p\bigr)=0. \] Here, \(c_j(d)\) are certain polynomial coefficients, \(H_k(p)\) is a combination of values \(_3F_2 (\lambda)\) for \(2\leq\lambda\leq p-1\), and \(\varepsilon_k(p)\) depends on \(k\) and \(p\) only; specifically, for \(p\equiv 1\bmod 4\) it depends on the decomposition of \(p\) as a sum of two squares. The formula implies that the generating function \(\sum_{k\geq 2\,\text{even} }\text{Tr}\,^{\text{new}}_{k-2j}(\Gamma_0(8),p)\cdot X^{k/2-1}\) is a rational function whose coefficients depend on \(p\) and the values \(_3F_2(\lambda)\), \(2\leq\lambda\leq p-1\). There are several applications: (1) The coefficients of the unique newform \(\eta(2z)^4\eta(4z)^4\) in \(S_4 (\Gamma_0(8))\) are expressed in terms of values \(_3F_2 (\lambda)\) or of \(_4F_3 (1)\); this was already known [S. Ahlgren, and K. Ono, J. Reine Angew. Math. 518, 187–212 (2000; Zbl 0940.33002)], and is equivalent to the assertion that a certain Calabi-Yau threefold is modular. (2) The coefficients of the unique newform \(\eta (4z)^4\cdot(\eta(z)^8+8\eta(4z)^8)\) in \(S_6(\Gamma_0 (8))\) can be expressed by \(_4F_3(1)\) and \(_6F_5(1)\), which was conjectured by M. Koike. (3) Ahlgren and 0no (loc. cit.) proved that \(\text{Tr}_4^{\text{new}} (\Gamma_0(8),p)\) is congruent to an Apéry number modulo \(p\); now the authors define generalized Apéry numbers and show that \(\text{Tr}_k^{\text{new}} (\Gamma_0(8),p)\) is congruent modulo \(p\) to a combination of generalized Apéry numbers for all even \(k\geq 4\). There are similar results for the traces \(\text{Tr}(\Gamma_0(N),p)\) for \(N=2\) and \(N=4\). We await further research on more general levels \(N\).

MSC:

11F25 Hecke-Petersson operators, differential operators (one variable)
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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