论文标题
衍生的Zeta功能，用于有限场上的曲线
Derived Zeta Functions for Curves over Finite Fields
论文作者
论文摘要
对于每个$（m+1）$ - 元组$ {\ bf n} _m =（n_0，n_1，\ ldots，n_m）$的正整数$ n} _m）}（s）$定义为曲线$ x $上的$ \ mathbb f_q $。此得出的ZETA函数满足标准的Zeta属性。特别是，类似于$ x/\ mathbb f_q $的artin zeta函数，此$ {\ bf n} _m $ $ $ $ x $ over $ x $ over $ \ mathbb f_q $的Zeta Zeta函数是$ 2G $ $ 2G $ poldonomial $ P_ $ t _ {{{\ bf n} _m} = q^{ - s \ s \ prod_ {k = 0}^mn_k} $ by $（1-t _ {{\ bf n} _mm} _mm}） n} _m}）t _ {{\ bf n} _m}^{g-1} $带有$ q _ {{{\ bf n} _m} = q^{\ prod_ {k = 0}^mn_k} $。实际上，我们有$$ \ begin {Aligned}＆\ wideHatζ_{x，\ Mathbb f_q}^{\，（{\ bf n} _ {m} _ {m}）}（s）= \ wideHat z____________________________________ n} _ {m}）}（t _ {{{\ bf n} _ {m}}）\\ =＆\ left（\ sum _ {\ sum _ {\ ell = 0}^{g-2} {g-2}α___ n} _ {m}）}（\ ell）\ big（t _ {{\ bf n} _ {m}}}^{\ ell-（g-1）}+q _ {{\ bf n} n} _ {m}}^{（g-1） - \ ell} \ big） +α_{x，x，\ mathbb f_q}^{（{\ bf n} _ {m} _ {m}）}}}（g-1）}（g-1）（g-1）（g-1））\ big）\ right）\ right） n} _ {m}} - 1） n} _ {m}}）（1- q _ {{{\ bf n} _ {m}} t _ {{{\ bf n} _ {m}}}} \\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ bf n} $ - f_q $。此外，当$ x $限制到椭圆曲线时，或者当$ {\ bf n} _m =（2,2，\ ldots 2）$时，确定的是$ {\ bf n} _m $ $ $ $ $ $ $ $ $ $ $ $ \ wide的所有zeros，则f_q}^{\，（（{\ bf n} _ {m}）}（s）$位于中心行$ \ re（s）= \ frac {1} {2} $上。此外，正式是积极的猜想，声称上述$ {\ bf n} _m $ $ $衍生的alpha和beta不变性都是严格的积极性。
For each $(m+1)$-tuple ${\bf n}_m=(n_0,n_1,\ldots,n_m)$ of positive integers, the ${\bf n}_m$-derived zeta function $\widehatζ_{X,\mathbb F_q}^{\,({\bf n}_m)}(s)$ is defined for a curve $X$ over $\mathbb F_q$. This derived zeta function satisfies standard zeta properties. In particular, similar to the Artin Zeta function of $X/\mathbb F_q$, this ${\bf n}_m$-derived Zeta function of $X$ over $\mathbb F_q$ is a ratio of a degree $2g$ polynomial $P_{X,\mathbb F_q}^{({\bf n}_m)}$ in $T_{{\bf n}_m}=q^{-s\prod_{k=0}^mn_k}$ by $(1-T_{{\bf n}_m})(1-q_{{\bf n}_m}T_{{\bf n}_m})T_{{\bf n}_m}^{g-1}$ with $q_{{\bf n}_m}=q^{\prod_{k=0}^mn_k}$. Indeed, we have $$\begin{aligned} &\widehat ζ_{X,\mathbb F_q}^{\,({\bf n}_{m})}(s)=\widehat Z_{X,\mathbb F_q}^{\,({\bf n}_{m})}(T_{{\bf n}_{m}})\\ =& \left(\sum_{\ell=0}^{g-2}α_{X,\mathbb F_q}^{({\bf n}_{m})}(\ell)\Big(T_{{\bf n}_{m}}^{\ell-(g-1)}+q_{{\bf n}_{m}}^{(g-1)-\ell}T_{{\bf n}_{m}}^{(g-1)-\ell}\Big) +α_{X,\mathbb F_q}^{({\bf n}_{m})}(g-1))\Big)\right)+\frac{(q_{{\bf n}_{m}}-1)T_{{\bf n}_{m}}β_{X,\mathbb F_q}^{({\bf n}_{m})}}{(1-T_{{\bf n}_{m}})(1-q_{{\bf n}_{m}}T_{{\bf n}_{m}})}\\ \end{aligned}$$ for some ${\bf n}_m$-derived alpha and beta invariants of $X/\mathbb F_q$. Furthermore, when $X$ restrict to an elliptic curve, or when ${\bf n}_m=(2,2,\ldots 2)$, established is the ${\bf n}_m$-derived Riemann hypothesis claiming that all zeros of $\widehat ζ_{X,\mathbb F_q}^{\,({\bf n}_{m})}(s)$ lie on the central line $\Re(s)=\frac{1}{2}$. In addition, formulated is the Positivity Conjecture claiming that the above ${\bf n}_m$-derived alpha and beta invariants are all strict positivity.
