Zeta functions of finite fields and the Selberg class
Volume 184 / 2018
Acta Arithmetica 184 (2018), 247-265 MSC: 11M41, 11G20, 11G25. DOI: 10.4064/aa170811-12-1 Published online: 26 July 2018
We analyze the relations between the zeta functions of smooth projective varieties over finite fields and the functions of degree $0$ from the extended Selberg class. In particular, denoting such functions by $\mathcal S_0^\sharp$, we first describe how to associate suitable local $L$-functions from $\mathcal S^\sharp_0$ to the varieties over a finite field. Then we show that, in a suitable sense and under a certain hypothesis, $\mathcal S_0^\sharp$ is generated by the local $L$-functions coming from curves.