How does the order of a pole of a zeta function indicate any geometric information? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:36:08Z http://mathoverflow.net/feeds/question/11207 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11207/how-does-the-order-of-a-pole-of-a-zeta-function-indicate-any-geometric-informatio How does the order of a pole of a zeta function indicate any geometric information? Yinbang Lin 2010-01-09T03:09:59Z 2010-02-28T08:50:18Z <p>Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic. </p> <p>Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the hypersurface defined by $-y_0^2+y_1^2+y_2^2+y_3^2=0$ in $\mathbb{P}^3$.<br /> If $-1$ is a square in $F_q$, the zeta function is </p> <p>$$Z(u)=\frac{1}{(1-uq^2)(1-uq)^2(1-u)}.$$</p> <p>It has a pole of order $2$ at $1/q$. If not, it's </p> <p>$$Z(u)=\frac{1}{(1-uq^2)(1-uq)(1+uq)(1-u)}.$$ </p> <p>It has a pole of order $1$ at $1/q$.</p> <blockquote> <p>How does orders of poles indicate any geometric information?</p> </blockquote> http://mathoverflow.net/questions/11207/how-does-the-order-of-a-pole-of-a-zeta-function-indicate-any-geometric-informatio/11215#11215 Answer by Felipe Voloch for How does the order of a pole of a zeta function indicate any geometric information? Felipe Voloch 2010-01-09T04:07:05Z 2010-01-09T04:22:05Z <p>For a smooth projective surface, the order of the pole at 1/q is conjectured to be the rank of the Neron-Severi group of the surface. That's a conjecture of Tate and is an analog of the Birch and Swinnerton-Dyer conjecture. Tate has formulated a more general conjecture for higher dimensional varieties too. For the case of quadrics, as in your example, these conjectures are known.</p> <p>Edit: maybe you don't want a fancy answer. In the first case, the quadric contains lines and in the second, it doesn't.</p> http://mathoverflow.net/questions/11207/how-does-the-order-of-a-pole-of-a-zeta-function-indicate-any-geometric-informatio/11233#11233 Answer by David Speyer for How does the order of a pole of a zeta function indicate any geometric information? David Speyer 2010-01-09T13:48:57Z 2010-01-09T13:48:57Z <p>This is an expository note filling in the background between Steven Sam's comment and Felipe Voloch's answer.</p> <p>If $X$ is a smooth projective variety, then the Weil conjectures (now theorems) describe the zeroes and poles of the zeta function in terms of the cohomology of $X$, and the action of Frobenius on it. In particular, the poles on the circle $|u|=1/q$ are the reciprocals of the eigenvalues of Frobenius acting on <code>$H^2(X, \mathbb{Q}_{\ell})$</code>. </p> <p>In your example, $H^2$ is two dimensional. Over the algebraic closure <code>$\overline{F_q}$</code>, your variety is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$. $H^2$ is spanned by the two classes <code>$\mathbb{P}^1 \times \{ \mbox{point} \}$</code> and <code>$\{ \mbox{point} \} \times \mathbb{P}^1$</code>.</p> <p>If $-1$ is a square, Frobenius acts on this two dimensional vector space by multiplication by $q$, so you get a double pole at $1/q$. If $-1$ is not a square, then Frobenius multiplies by $q$ and switches the two generators. So the eigenvalues are $q$ and $-q$.</p>