##
**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 \(N\).

Reviewer: Günter Köhler (Würzburg)