# Related Forms for the Riemann Hypothesis over Finite Fields

There are several formulations and consequences of the Riemann Hypothesis for Curves over Finite Fields. I am interested in the logical implications between those, and in elementary (as possible) proofs\references for those implications (which are surely known to the experts).

I will state the formulations\consequences. Let $\mathbb{F}_q$ be the finite field with $q$ elements.

1. Curves: Let $C/\overline{\mathbb{F}_q}$ be a smooth, projective algebraic curve defined over $\mathbb{F}_q$. Let $\zeta_C(s)$ be the zeta function of $C$, defined as $$\zeta_C(s) = \exp(\sum_{n \ge 1} \frac{N_m}{m}q^{-ms}),$$ where $N_m$ is the number of points of $C$ defined over the degree $m$ extension $\mathbb{F}_{q^m}$ of $\mathbb{F}_q$. RH: All the zeros of $\zeta_C(s)$ lie on the line $\Re(s)=\frac{1}{2}$.
2. Field Extensions: Let $K/\mathbb{F}_q$ be a function field with constant field $\mathbb{F}_q$. Let $\zeta_K(s)$ be the zeta function of $K$, defined as follows: $$\zeta_K(s) = \sum_{A \ge 0} (NA)^{-s},$$ where the summation is over all effective divisors $A$ of $K$, and $NA=q^{\deg A}$. RH implies: All the zeros of $\zeta_K(s)$ lie on the line $\Re(s)=\frac{1}{2}$.
3. Rings of Integers (Dedekind zeta functions): Let $K/\mathbb{F}_q(T)$ be a field extension of finite degree. Let $O_K$ be the integral closure of $\mathbb{F}_q[T]$ in $K$. Let $\zeta_{O_K}(s)$ be the zeta function of $O_K$, defined as follows: $$\zeta_{O_K}(s) = \sum_{I \ge 0} (NI)^{-s},$$ where the summation is over all ideals $I$ of $O_K$, and $NI=|O_K/I|$. RH implies: All the zeros of $\zeta_{O_K}(s)$ lie on the line $\Re(s)=\frac{1}{2}$.
4. Characters (L-functions): Let $\chi:\mathbb{F}_q[T] \to \mathbb{C}$ be a generalized Dirichlet character. Let $$L(s,\chi) = \sum_{f \in \mathbb{F}_q[T], f \text{ monic}} \chi(f) |f|^{-s}$$ be its L-function, where $|f|=|\mathbb{F}_q[T]/f|$. RH implies: All the zeros of $L(s,\chi)$ lie on the line $\Re(s) = \frac{1}{2}$.

I believe the implications are: $1 \leftrightarrow 2 \leftrightarrow 3 \implies 4$, although I cannot show this rigorously. Are these the only implications, and are they correct?

I have some hand-waving arguments that partially explain the implications, but these are not proofs, and I am not satisfied with them:

• I am most comfortable with Variant 2. It has an elementary proof due to Stepanov-Bombieri, found in the appendix "Number Theory in Function Fields" by Michael Rosen.
• Morally, Variant 1 and Variant 2 are equivalent, since one can associate a function field to any curve, and vice versa.
• The only difference between Variant 2 and Variant 3 seems to be the contribution of the prime at infinity, which only contributes a pole ($s=1$).
• Variant 3 implies Variant 4, as follows: Note $L(s,\chi)$ is a polynomial. Construct an abelian extension $K$ of $\mathbb{F}_q(T)$ whose set $S$ of "associated Dirichlet characters" contains $\chi$. In that case, $\zeta_{O_K}(s) = \prod_{\chi' \in S} L(s,\chi')$. RH for $\zeta_{O_K}(s)$ implies $RH$ for $L(s,\chi)$. The hard part is the construction of the abelian extension. Is there an alternative argument (or an easy way to see the existence of such $K$?)
• maybe you should try asking this on MathOverflow. – Peter Humphries Feb 29 '16 at 4:53
• In case it may be useful OP asked in mathoverflow as suggested in here – Daniel D. Nov 26 '19 at 2:27