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Motivation for zeta perform of an algebraic selection Answer

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Motivation for zeta perform of an algebraic selection

If $pi:Xrightarrow Spec(mathbb{Z})$ is a strategy of finite kind over the ring of integers it’s possible you’ll outline the Hasse-Weil L-function $L(X,s)$ of $X$ as follows:

L0. Definition: $L(X,s):= prod_{x in X^{cl}}frac{1}{1-N(x)^{-s}}$

the place $N(x)$ is the quantity of components within the residue bailiwick $kappa(x)$ of $x$. Since $x$ is a closed level it follows $kappa(x)$ is a finite bailiwick.

Example: If $X:=Spec(mathcal{O}_K)$ with $Okay$ a quantity bailiwick it follows $L(X,s)$ is the Dedekind L-function.

In common there’s the next method:

L1. $L(X,s) = prod_{0neq pin mathbb{Z}} L(X_p,s)$

the place $X_p:=pi^{-1}(p)$ is the fiber of $pi$ on the nonzero prime $p$. The fiber $X_p$ is a strategy of finite kind over $mathbb{F}_p:=mathbb{Z}/(p)mathbb{Z}$, and there’s an equality

L2. $L(X_p,s)=Z(X_p, p^{-s})$

the place $Z(X_p,t)$ is the Weil zeta perform of $X_p$ as outlined in Hartshorne’s (HH) classical bespeak (Appendix C, rehearse 5.4). Hence your perform $zeta_{V,p}(s)$ equals by ex HH.C.5.4 the perform $Z(V, p^{-s})$, the place $Z(V,t)$ is outlined on web page 450 in HH.

Hence you get the method

L3. $L(X,s)= prod_{pin mathbb{Z}} Z(X_p, p^{-s})$.

Hence the perform in L0 provides one unified generalization of the perform you outline and the classical Dedekind L-function. The Hasse-Weil L-function $L(X,s)$ is conjectured to answer the next situation:

L4. $ord_{s=ok}(L(X,s))=chi(X,ok)$

See. web page 501, C2.2 within the following ICM1983 continuing:

https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1983.1/ICM1983.1.ocr.pdf

the place $chi(X,ok)$ is the $ok$‘th Euler attribute of $X$.

By definition

L5. $chi(X,ok):= sum_{min mathbb{Z}} dim_{mathbb{Q}}(-1)^{m+1}(operatorname{Okay}_{m}'(X)_{(ok)})$

When $ok=1$ and $X:=A$ is an abelian strategy, Conjecture L4 is “a version” of the BSD surmise for $A$:

BSD: $ord_{s=1}(L(A,s))= chi(A,1)$.

If $E^*$ is an elliptic round over a quantity bailiwick and $E$ is the Neron mannequin of $E^*$ it’s conjectured that

L6. $ord_{s=1}(L(E,s))=chi(E,1)$.

The classical BSD conjeture is formulated for an elliptic round $F$ over $mathbb{Q}$ utilizing the group of rational factors $F(mathbb{Q})$ and an L-function differing from the L-function in L6.
At the Clay Math Institute dwelling web page you discover some motivation for the seek of BSD surmise within the “official problem description”:

“In conclusion, although there has been some success in the last fifty years in limiting the number of rational points on varieties, there are still almost no methods for finding such points. It is to be hoped that a proof of the Birch and Swinnerton-Dyer conjecture will give some insight concerning this general problem.”

http://www.claymath.org/websites/default/recordsdata/birchswin.pdf

Since your zeta perform is said to the Hasse-Weil L-function (by way of the product method L1) and rational poins on algebraic varieties, it appears this offers motivation for this seek. In reality the issue description says the next:

“Since the original conjecture was stated, much more elaborate conjectures concerning special values of L-functions have appeared, due to Tate,Lichtenbaum, Deligne, Bloch, Beilinson and others, see [21], [3] and [2]. In particular, these relate the ranks of groups of algebraic cycles to the order of vanishing (or the order of poles) of suitable L-functions.”

It appears he’s talking about Conjecture L4 above. Note that for a strategy of finite kind over a bailiwick there’s a relation between algebraic Okay-theory and Bloch’s Chow teams (with rational coefficients), however this relation isn’t established over the ring of integers.

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