## 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)

### Citations:

Zbl 0629.12017; Zbl 0784.11057; Zbl 0910.11054; Zbl 0940.33002
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