# Lecture 1 ## References - - [@milneLEC], [@milne_2017], [@freitag_kiehl_2013], [@katz] ## Intro Prerequisites: - Homological Algebra - Abelian Categories - Derived Functors - Spectral Sequences (just exposure!) - Sheaf theory and sheaf cohomology - Schemes (Hartshorne II and III) Outline/Goals: - Basics of etale cohomology - Etale morphism - Grothendieck topologies - The etale topology - Etale cohomology and the basis theorems - Etale cohomology of curves - Comparison theorems to singular cohomology - Focused on the case where coefficients are a constructible sheaf. - Prove the Weil Conjectures (more than one proof) - Proving the Riemann Hypothesis for varieties over finite fields > One of the greatest pieces of 20th century mathematics! - Topics - Weil 2 (Strengthening of RH, used in practice) - Formality of algebraic varieties (topological features unique to varieties) - Other things (monodromy, refer to Katz' AWS notes) ## What is Etale Cohomology? Suppose $X/\CC$ is a quasiprojective variety: a finite type separated integral $\CC\dash$scheme. If you take the complex points, it naturally has the structure of a complex analytic space $X(\CC)^{\text{an}}$: you can give it the Euclidean topology, which is much finer than the Zariski topology. For a nice topological space, we can associate the singular cohomology $H^i(X(\CC)^{\text{an}}, \ZZ)$, which satisfies several nice properties: - Finitely generated $\ZZ\dash$modules - Extra Hodge structure when tensored up to $\CC$ (same as $\CC$ coefficients) - Cycle classes (i.e. associate to a subvariety a class in cohomology) Goal of etale cohomology: do something similar for much more general "nice" schemes. Note that some of these properties are special to complex varieties. (E.g. finitely generated: not true for a random topological space.) We'll associate to $X$ a "nice scheme" $\rightsquigarrow H^i(X_{\text{et}}, \ZZ/\ell^n\ZZ)$. Take the inverse limit over all $n$ to obtain the $\ell\dash$adic cohomology $H^i(X_{\text{et}}, \ZZ_\ell)$. You can tensor with $\QQ$ to get something with $\QQ_\ell$ coefficients. And as in singular cohomology, you can a "twisted coefficient system". :::{.example title="?"} What are some nice schemes? - $X = \spec \OO_k$, the ring of integers over a number field. - $X$ a variety over an algebraically closed field - Typical, most analogous to taking a variety over $\CC$. - $X$ a variety over a non-algebraically closed field ::: Some comparisons between the last two cases: - For $\CC\dash$ variety, $H^i_{\text{sing}}$ will vanish above $i=2d$. - Over a finite field, $H^i$ will vanish for $i>2d+1$ but generally not vanish for $i=2d+1$. In good situations, these are finitely generated $\ZZ/\ell^n\ZZ\dash$modules, have Mayer-Vietoris and excision sequences, spectral sequences, etc. Related invariants: for a scheme with a geometric[^what_is_a_geom_point] point: $(X, \bar x) \rightsquigarrow \pi_1^{\text{étale}}(X, \bar x)$, which is a profinite topological group. [^what_is_a_geom_point]: A **geometric point** is a map from $\spec X$ to an algebraically closed field. :::{.remark} More invariants beyond the scope of this course: - Higher homotopy groups - Homotopy type (equivalence class of spaces) So we want homotopy-theoretic invariants for varieties. ::: :::{.remark} This cohomology theory is necessarily weird! The following theorem explains why. The slogan: there is no cohomology theory with $\QQ$ coefficients. ::: :::{.theorem title="Serre"} There does not exists a cohomology theory for schemes over $\bar{\FF}_q$ with the following properties: 1. Functorial 2. Satisfies the Kunneth formula 3. For $E$ an elliptic curve, $H^1(E) = \QQ^2$. ::: :::{.proof} Take $E$ to be a supersingular elliptic curve. Then $\Endo(E) \tensor \QQ$ is a quaternion algebra, and use the fact that there are no algebra morphisms $R\to \mat_{2\times 2}(\QQ)$. ::: :::{.exercise} Functoriality and Kunneth implies that $\Endo(E)\actson E$ yields an action on $H^1(E)$, which is precisely an algebra morphism $\Endo(E) \to \mat_{2\by 2}(\QQ)$, a contradiction. The content here: the sum of two endomorphisms act via their sum on $H^1$. ::: :::{.exercise} Prove the same thing for $\QQ_p$ coefficients, where $p$ divides the characteristic of the ground field. Proof the same, just need to know what quaternion algebras show up. ::: This forces using some funky type of coefficients. ## What are the Weil Conjectures? Suppose $X/\FF_q$ is a variety, then \[ \zeta_X(t) = \exp{\sum_{n>0} { {\abs{X(\FF_{q^n})} \over n} t^n } } .\] :::{.remark} \envlist - $\dd{}{t} \log \zeta_X(t)$ is an ordinary generating function for the number of rational points. - Slogan: locations of zeros and poles of a meromorphic function control the growth rate of the coefficients of the Taylor series of the logarithmic derivative. ::: :::{.exercise} Make this slogan precise for rational functions, i.e. ratios of two polynomials. ::: :::{.theorem title="The Weil Conjectures"} \envlist 1. $\zeta_x(t)$ is a rational function. 2. (Functional equation) For $X$ smooth and proper \[ \zeta_X(q^{-n} t\inv) = \pm q^{nE \over 2} t^E \zeta_X(t) .\] 3. (RH) All roots and poles of $\zeta_X(t)$ have absolute value $q^{i\over 2}$ with $i\in \ZZ$, and these are equal to the $i$th Betti numbers if $X$ lifts to characteristic zero.[^generalize_betti_numbers] [^generalize_betti_numbers]: Note that we'll generalize Betti numbers so this makes sense in general. ::: :::{.remark} These are all theorems! The proofs: 1. Dwork, using $p\dash$adic methods. Proof here will follow from the fact that $H^i_{\text{étale} }$ are finite-dimensional. Related to the **Lefschetz Trace Formula**, and is how Grothendieck thought about it. 2. Grothendieck, follows from some version of Poincaré duality. 3. (and 4) Deligne. ::: ### Euler Product Let $\abs X$ denote the closed points of $X$, then there is an Euler product: \[ \zeta_X(q^{-n} t\inv) = \pm q^{nE \over 2} t^E \zeta_X(t) &= \prod_{x\in \abs{X}} \exp{t^{\deg (x)} + {t^{2\deg (x)} \over 2} + \cdots} \\ &= \prod_{x\in \abs X} \exp{-\log(1-t^{\deg(x)})} \\ &= \prod_{x\in \abs X} {1 \over 1 - t^{\deg(x)}} .\] :::{.exercise} Check the first equality. If you have a point of $\deg(x) = n$, how many $\FF_{q^n}$ points does this contribute? I.e., how many maps are there $\spec(\FF_{q^n}) \to \spec(\FF_{q^n})$ over $\FF_q$? There are exactly $n$: it's $\gal(\FF_{q^n} / \FF_q)$. But then division by $n$ drops this contribution to one. ::: We can keep going by expanding and multiplying out the product: \[ \prod_{x\in \abs X} {1 \over 1 - t^{\deg(x)}} &= \prod_{x\in \abs X} (1 + t^{\deg(x)} + t^{2 \deg(x)}) \\ &= \sum_{n\geq 0} \qty{\text{\# of Galois-stable subset of $X(\bar \FF_{q^n})$ of size $n$}}t^n .\] Why? If you have a degree $x$ point, it contributes a stable subset of size $x$: namely the $\FF_{q^n}$ points of $\FF_{q^n}$. Taking Galois orbits like that correspond to multiplying this product. But these are the points of some algebraic variety: \[ \cdots = \sum_{n\geq 0} \abs{\sym^n(X)(\FF_q)} t^n ,\] where $\sym^n(X) \da X^n/\Sigma_n$, the action of the symmetric group. Why is that? A $\bar \FF_q$ point of $\sym^n(X)$ is an unordered $n\dash$tuple of $\bar \FF_q$ points without an ordering, and asking them to be Galois stable is the same as saying that they are an $\FF_q$ point. :::{.theorem title="First Weil Conjecture for Curves"} For $X$ a smooth proper curve over $\FF_q$, $\zeta_X(t)$ is rational. ::: :::{.proof} :::{.claim} There is a set map \[ \sym^n X &\to \pic^n X \\ D &\mapsto \OO(D) ,\] where here the divisor is an $n\dash$tuple of points. ::: What are the fibers over a line bundle $\OO(D)$? A linear system, i.e. the projectivization of global sections $\PP \Gamma(X, \OO(D))$. What is the expected dimension? To compute the dimension of the space of line bundles on a curve, use Riemann-Roch: \[ \dim \PP\Gamma(X, \OO(D)) = \deg(D) + 1 - g + \dim H^1(X, \OO(D)) - 1 .\] where the last $-1$ comes from the fact that this is a projective space. :::{.claim} If $\deg(D) = 2g-2$, then $H^1(X, \OO(D)) = 0$. ::: This is because it's dual to $H^0(X, \OO(K-D))\dual$, but this has negative degree and a line bundle of negative degree can never have sections.[^check_this_1] Thus the fibers are isomorphic to $\PP^{n-g}$ for $n>2g-2$. Now make a reduction[^exc_justify_reduction_1] and without loss of generality, assume $X(\FF_q) \neq \emptyset$. In this case, $\pic^n(X) \cong \pic^{n+1}(X)$ for all $n$, since you can take $\OO(P)$ for $P$ a point, a degree 1 line bundle, and tensor with it. It's an isomorphism because you can tensor with the dual bundle to go back. Thus for all $n>2g-2$, \[ \abs{\sym^n(X)(\FF_q)} = \abs{\PP^{n-g}(\FF_q)} \cdot \abs{\pic^n(X)(\FF_q)} = \abs{\PP^{n-g}(\FF_q)} \cdot \abs{\pic^0(X)(\FF_q)} .\] Thus $\zeta_X(t)$ is a polynomial plus $\sum_{n>2g-2} \abs{\pic^n(X)(\FF_q)}\qty{1+q+q^2+\cdots+q^{n-g}}t^n$. [^exc_justify_reduction_1]: Exercise: justify why the reduction is valid. [^check_this_1]: You should check to make sure you know why this is true! :::{.exercise} Show that this is a rational function using the formula for a geometric series. ::: ::: :::{.theorem title="Functional Equation"} The functional equation in the case of curves: \[ \zeta_X(q^{-1} t^{-1} ) = \pm q^{2-2g \over 2} t^{2-2g} \zeta_X(t) .\] ::: :::{.exercise title="Important"} Where it comes from in terms of $\sym^n$: Serre duality. ::: We'll show the RH later. :::{.theorem title="Dwork"} Suppose $X/\FF_q$ is any variety, then $\zeta_X(t)$ is rational function. ::: This was roughly known to Weil, hinted at in original paper :::{.proof title="Grothendieck"} Idea: take Frobenius (intentionally vague, arithmetic vs geometric vs ...) $F:X\to X$, then $X(\FF_q)$ are the fixed points of $F$ acting on $X_{\bar \FF_q}$, and the $\FF_{q^n}$ points are the fixed points of $F^n$. Uses the Lefschetz fixed point formula, which will say for $\ell\neq \ch(\FF_q)$,[^compact_supported_cohomology] \[ \abs{X(\FF_{q^n})} = \sum_{i=0}^{2\dim(X)} (-1)^i \tr(F^n) H^i_c(X_{\FF_q}, \QQ_\ell) .\] [^compact_supported_cohomology]: Here $H^i_c$ is compactly supported cohomology, we'll define this later in the course. :::{.lemma} \[ \exp{\sum {\tr(F^n) \over n}t^n }\quad\text{is rational} .\] ::: This lemma implies the result, because if you plug the trace formula into the zeta function, you'll get an alternating product $f \cdots {1\over g} \cdot h \cdot {1\over j} \cdots$ of functions of the form in the lemma, which is still rational. ::: :::{.proof title="Of Lemma"} It suffices to treat the case $\dim(V) = 1$, because otherwise you can just write this as a sum of powers of eigenvalues. Then you have a scalar matrix, so you obtain \[ \exp{ \sum {\alpha^n \over n} t^n} = \exp{ -\log(1 - \alpha t)} = {1 \over 1-\alpha t} ,\] which is rational. ::: This proves rationality, contingent on 1. The Lefschetz fixed point formula 2. The finite dimensionality of etale cohomology :::{.exercise} Try to figure out how Poincaré duality should give the functional equation. *(Hint: try the lemma on a vector space where $F$ scales a bilinear form.)* ::: :::{.theorem title="Serre, Kahler Analog"} Suppose $X/\CC$ is a smooth projective variety and $[H] \in H^2(X(\CC), \CC)$ is a hyperplane class (corresponds to intersection of generic hyperplane or the first Chern class of an ample line bundle). Suppose $F:X\to X$ is an endomorphism such that $f^*[H] = q[H]$ for some $q\in \ZZ_{\geq 1}$. Define \[ L(F^n) \definedas \sum_{i=0}^{2\dim(X)} (-1)^i \tr\qty{ F^n \st H^i(X_{\FF_q}, \QQ_\ell)} .\] and \[ \zeta_{X, F}(t) \da \exp{\sum_{n=1}^\infty {L(F^n) \over n}t^n } .\] Then $\zeta_{X, F}(t)$ satisfies the RH: the zeros and poles are of absolute value $q^{i\over 2}$. Equivalently, the eigenvalues $\lambda$ of $F^n$ acting on $H^i(X, \CC)$ all satisfy $\abs{\lambda} = q^{i\over 2}$. ::: Next time, to review - Étale morphisms - Definition of a site asdsadsadsa dasdsadsa dasdsda