Following up this thread: $L$-function of an elliptic curve and isomorphism class
I'd like to ask some more questions for the case of smooth projective curves $C$ over $\mathbb Q$
To be more precise, take this definition ( https://webusers.imj-prg.fr/~marc.hindry/Notes_rev_Brasilia.pdf, (2.2)):
$$L(\rho,s)=\prod_{\mathfrak p}\det(1-\rho(\mathrm{Frob}_p)p^{-s}\mid V^{I_{\mathfrak p}})^{-1}$$ where $\mathfrak p$ runs over the primes of $\mathbb Q$, or more generally: the number field $K$. And $V=H^1_{et}(C,\mathbb Q_\ell)$.
- Is there a similar statement for curves of higher genus? I.e. if two $L$-functions $L(C,s)$ and $L(C',s)$ coincide, what can be said about $C$ and $C'$?
- Are there further interesting properties that can be read off the $L$-function $L(C,s)$, and not the curve $C$ itself?
- Assuming the Generalized Riemann Hypothesis holds for $L$-functions of curves over $\mathbb Q$, what does that imply about the curve?
Best,
Dan
Let $A:=\mathbb{Z}$ be the ring of integers and let $S$ be a scheme of finite type over $A$. If $\pi:X \rightarrow S$ is a scheme of finite type over $S$, you may define the Hasse-Weil L-function $L(X,s):=\prod_{x \in X^{cl}} \frac{1}{1-N(x)^{-s}}$. Here $N(x)$ is the number of elements in in the residue field of $x$. Since $x$ by definition is a closed point it follows $N(x)$ is an integer. If $\pi':Y\rightarrow S$ is another scheme of finite type and $X_t\cong Y_t$ are isomorphic for any closed point $t \in S$, it follows $L(X,s)=L(Y,s)$. In particular if $E,F$ are vector or projective bundles of the same rank over $S$ it follows $L(E,s)=L(F,s)$. Hence if $E$ and $F$ have the same L-function this does not imply that $E$ and $F$ are isomorphic. Hence if $S=C$ is a "curve" it follows vector or projective bundles over $C$ are not determined by their Hasse-Weil L-functions.