--- date: 2023-01-12 11:05 aliases: ["Winter 2022 Reading Group Talk Final"] --- Last modified: `=this.file.mday` --- - Tags: - #my/talks - Refs: - #todo/add-references - Links: - #todo/create-links --- # Motivation - Notation - Overview - Classical theory and analogues - Bernoulli numbers - Hilbert's 12th problem --- ## Bernoulli numbers are cool - Define them by $$\frac{z}{e^z-1}= {z\over 2} \qty{\coth\qty{z\over 2} -1 } = \sum_{n=0}^{\infty} B_n {z^n\over n!} \leadsto \tv{1, -{1\over 2}, {1\over 6}, {1\over 30}, \cdots}$$ - Show up in (normalized) Eisenstein series $$E_{2 k}(q)=1-\frac{4 k}{B_{2 k}} \sum_{n \geq 1} \sigma_{2 k-1}(n) q^n$$ - Show up in stable homotopy theory: $$\operatorname{Im}\left(J_{4 k-1}\right) \simeq \mathbf{Z} / D_{2 k}\qquad D_{2 k}=\text { the denominator of } B_{2 k} / 4 k \text {. }$$ - The $J$-homomorphism is a group homomorphism $J_{k, n}: \pi_k(\mathrm{SO}(n)) \rightarrow \pi_{n+k}\left(S^n\right)$. This map passes to a stable $J$-homomorphism $$J_k: \pi_k(\SO) \rightarrow \pi_k(\SS)$$ which comes from framed cobordism. - Congruences of $E_{2k}$ and images of $J$ explained by elliptic cohomology and topological modular forms (topics in chromatic homotopy). ## Bernoulli numbers in zeta functions - Recall $$\zeta(s) \da \sum_{n\geq 0} n^{-s} = \prod_{p} {1\over 1-p^{-s}}, \qquad \Re(s) > 1$$ the Dirichlet $L\dash$function $L(\chi, s)$ for the trivial character $\chi(n) = 1$. This diverges in $0 < \Re(s) < 1$ but has a meromorphic continuation to all of $\CC$ by functional and integral equations. - Special even values of zeta: for $n\geq 1$, the Bernoulli identity $$\zeta(2k)=-\frac{(2 \pi i)^{2k}}{2 (2k) !} B_{2k}$$ - Note the rational factor $1/2$, the algebraic factors $i$, and transcendental factors $\pi^{2k}$, the factorial. - E.g. $\zeta(2) = \pi^2/6$ (Basel problem) - Key ingredient in the proof: the Weierstrass product factorization $$e^z=\sum_{n=0}^{\infty} \frac{z^n}{n !}, \qquad \sinh(z) = \frac{e^z-e^{-z}}{2}=\pi z \prod_{n=1}^{\infty}\left(1+\frac{z^2}{n^2}\right)$$ ## Carlitz's Analogues Define $[n] = T^{q^n} - T$. | Number Fields/Arithmetic | Function Fields/Geometric | |:---------------------------------------------------------------------------------------------------------------|:----------------------------------------------------------------------------------------------------------------------------------------------------| | $K/\QQ$ a number field | $X$ a smooth projective geometrically integral curve over $\FF_q$ | | $$\CC\units, \GG_m$$ | The Carlitz module $$C: \FF_q[T] \to \CC_\infty\ts{\tau}, C(t) =T + \tau$$ | | $\ZZ$ | $A = \FF_q[T]$ | | $\ZZ_{> 0}$ | $A_+$ monic polynomials in $A$ | | $\QQ$ | $K = \FF_q(T)$ | | $\RR$ | $K_\infty$ or $\bar K_\infty$? | | $\CC$ | $\CC_\infty = \widehat{\bar K_\infty}$ its analytic closure | | $$\zeta(s) = \sum_{n\in \ZZ_{\gt 0}} n^{-s}$$ | $$\zeta_C(s) = \sum_{a\in A_+} a^{-s}$$ | | $n!$ | $$D_0 = 1,\qquad D_n = (T^{q^n}-T)D_{n-1}^q$$ | | $\pi$ | $$\pi_C=\prod_{n=1}^{\infty}\left(1-\frac{[n]_C}{[n+1]_C}\right)$$ | | $e^z$ and $\Lambda = 2\pi i \ZZ$ | $$\exp_\Lambda(z) \da \prod_{\lambda \in \Lambda} \qty{1 - {z\over \lambda}}$$ | | $$2\pi i \ZZ \injects \CC \surjectsvia{e^z} \CC$$ | $$\pi_C\, i_C \, A \injects \CC_\infty \surjectsvia{e_C(z)} \CC_\infty$$ | | $2 = \size(\ZZ\units)$ | $q-1 = \size A\units$ | | $i = \sqrt{-1}$ | $$i_C=\left(-[1]_C\right)^{1 /(q-1)}$$ | | $e^z = \sum_{n\geq 0} {z^n\over n!} \in \QQ\fps{z}$ | $$e_C(z) = \sum_{n\geq 0} {(z^q)^n \over D_n} \in K\fps{z}$$ | | $$\sinh(z) = \frac{e^z-e^{-z}}{2}=\pi z \prod_{n=1}^{\infty}\left(1+\frac{z^2}{n^2}\right)$$ | $$e_C(z)=z \prod_{\alpha \in \pi_C\, i_C\, A\smz}\left(1-\frac{z}{\alpha}\right)$$ | | $${z\over e^z-1} = \sum_{n\geq 0} \tilde B_n z^n,\,\, \tilde B_n = n! B_n$$ | $${z \over e_C(z)} = \sum_{m=0}^{\infty} \mathrm{B C}_m z^m$$ | | $$\zeta(k)=-\frac{( 2\pi i)^{k}}{2} \tilde B_{k},\,\, 2\mid k$$ | $$\zeta_C(k) = - {(\pi_C i_C)^k \over q-1}\mathrm{BC}_k, \,\, q-1\mid k$$ | | $\zeta(-k) \in \QQ$ for $k\in \ZZ_{>0}$ | $\zeta_C(-m)\in A$ for $m \in \ZZ_{\gt 0}$ (Goss 79) | | $\zeta(-k) = 0$ for $k\equiv 0\mod 2$ | $\zeta(-m) = 0$ when $k\equiv 0 \mod (q-1)$ (Goss 79) | | $\pi$ is transcendental over $\QQ$ | $\pi_C$ is transcendental over $K$ (Wade, 40s) | | Conjecture: $\zeta(2n+1)$ is transcendental over $\QQ$ for all $n\geq 1$ | Theorem (Yu, 90s): $\zeta_C(n)$ is transcendental over $K$ for all $n\geq 1$ | | Conjecture: all zeros of a shifted/rotate zeta function $\tilde \zeta(s)$ are simple and on the line $\RR$ | Theorem (Wan 96, Thakur-Diaz-Vargas 96, Sheats 98): all zeros of $\zeta_{\mathrm{CG}}(x, -y)$ are on the "line" $K_\infty \da \FF_q\fls{T\inv}$ | | Elliptic curves | Drinfeld modules of rank 2 | | Abelian varieties | Drinfeld modules of rank $r\gt 2$ | | $\Endo_{\FF_q}(\GG_a(\CC))$ | $\CC_\infty\ts{\tau}$ | - Follow-up work proves continuations for analogs of Dedekind and Artin zeta functions. cohomologial proofs of analytic behavior, formulas for special values in analogy to BSD... ## Motivations from uniformization - Idea: formulate notions of lattices and exponentials to function fields in positive characteristic - Carlitz: studied explicit class field theory for $\FF_q[T]$ using "lattices" - Drinfeld: higher rank lattices (Drinfeld modules, give an analog of CM theory for function fields) - Anderson: higher dimensional Drinfeld modules ($T\dash$modules) and a motivic theory ($T\dash$motives). - Shtukas are generalizations, used to prove Langlands for $\GL_2(K)$, $K$ a function field. Laurent Lafforgue generalizes to $\GL_n(K)$ in the ff setting using moduli stacks of shtukas. - Need a bijection between (cuspidal) automorphic reps of $\GL_n(K)$ and (certain) Galois reps. - Drinfeld finds the reps in $H^*(\mathrm{Shtuka}^{r}; \ZZladic)$ (l-adic cohomology), a moduli stack of rank $r$ shtukas. - Shtukas are like vector bundles, defined by cospans. # Definitions - Motivating Drinfeld modules: - Start with $K\subseteq \CC \in \NF$ and some $d\geq 1$. - Consider $\Lambda \in \mods{\OO_K}$ of rank $d$. - Try to consider $\CC/\Lambda$ as an algebraic group: an *elliptic $\OO_K\dash$module*. - E.g. $\GG_m$ is an elliptic $\ZZ\dash$module of rank 1, - An elliptic curve is an elliptic $\ZZ\dash$module of rank 2. - This is all we see for $K\in \NF$ since one must have $d [K: \QQ]\leq 2$ forces imaginary quadratic - **Idea**: can we form elliptic modules over general global fields? - Replace $\CC$ with $\CC_v$, the Cauchy completion of an algebraic closure $\bar{K_v}$. - Look for $\OO_{K, S}\dash$submodules of $\CC_v$ -- these are discrete iff $S = \ts{v}$. Note $$\mathcal{O}_{K, S}=\left\{\left.x \in K|| x\right|_v \leq 1, v \notin S\right\} .$$ - Note that for number fields, this forces $S = \ts{\infty}$, so only 1 archimedean place, so either $K=\QQ$ or $K$ is an imaginary quadratic, forcing only rank 1 or 2 submodules. - More freedom on the function field side! - **Hilbert's 12th problem**: can special values of transcendental functions be used for explicit class field theory? - I.e. for a number field $K$, can on construct abelian extensions, similar to how $f(z) = \exp(2\pi i z)$ does for $K= \QQ$? - $d=1$: $\QQ(\exp(2\pi i n / m))$ is the ray class field $H_\QQ(m)$ for any $n$ coprime to $m$, so true for $K = \QQ$ (essentially Kronecker-Weber). - $d=2$: $K\in \NF$ imaginary quadratic, the answer is yes: identify the modulus with a nonzero ideal $\mfm\in \OO_K$, get the Hilbert class field $K_1 = K[j(E)]$ where $E$ is any elliptic curve with CM by $\OO_K$, and $K_\mfm = K_1[E[\mfm]]$ is given by adjoining the $\mfm\dash$torsion of one (or all) such curves. (Here the profinite completion of $C_\mfm$ is $\Gal(K_\mfm/K)$ where $C_\mfm$ is the ray class group.) - Drinfeld module - **Setup**: - $X$ a smooth projective geometrically irreducible curve over $\FF_q$ - $\infty$ a fixed place - $k$ the function field of $X$ (replaces $\QQ$) - $K\da K_\infty$ its completion (replaces $\RR$) - $A$ the functions regular away from $\infty$, e.g. $A = \FF_q[T]$, think of as $H^0(X\smts{\infty}; \OO_X)$. (replaces $\ZZ$) - $L$ is an $A\dash$field: a field $L$ with a morphism $i: A\to L$ and $\mfp \da \ker i$ is its characteristic. - $\iota: A\to L$ a fixed morphism in $\algs{\FF_q}$. - $L/\FF_q$ be an arbitrary extension - Definition (Drinfeld, 74): A **Drinfeld module of rank $r$ over $L$** is a morphism $$\phi: A\to L\ts{\tau} \qquad\in \algs{\FF_q}, \qquad \phi(T) = \iota(T) + \bigo(\tau)$$ - ![](attachments/2023-01-12diag.png) - Here $r$ is the highest power of $\tau$ that appears in $\bigo(\tau)$. - $i: A\to L$ is a ring morphism, so $L\in\algs{A}\intersect \Field$. - Think of $L\ts{\tau} \subseteq \Endo_{\FF_q}(\GG_a/L)$ ($\FF_q\dash$linear automorphisms of the additive group of $L$) - This induces an $A\dash$module structure on $L$ and identify $\phi$ with this module structure. - Geometrically, $\phi$ is $\GG_a/L$, but think of it as a pair $(\phi, \GG_a)$ where $\phi: A\to \Endo_L(\GG_a)$. - Alternatively: $\phi: A\actson \Endo\qty{\GG_a(L)}$. - Note that $\dd{}{x} \sum a_i x^{q^i} = a_0$, so this is like $A$ acting on the "tangent space" of $L$ by $i(a)$. - Compare to $E/k$ an elliptic curve, which is a $\zmod$ where $n\actson E(\QQ)$ by the group structure but $n\actson \T E$ by the usual multiplication - Examples - Example of a Drinfeld module: take $A = \FF_2[x,y]/\gens{y^2+y+x^3+x+1}$, then $A = H^0(X\smts\infty; \OO_X)$ for an elliptic curve $X$ over $\FF_2$. There are degree 1 Drinfeld modules: $$\begin{aligned} & \phi_x=x+\left(x^2+x\right) \tau+\tau^2 \\ & \phi_y=y+\left(y^2+y\right) \tau+x\left(y^2+y\right) \tau^2+\tau^3 \end{aligned}$$ - (Generic) characteristic - $\phi$ has **generic characteristic** if $\iota: A\to L$ injects, otherwise $\phi$ has **characteristic $\mfp$** where $\mfp \da \ker \iota$ - **Defined** over subfields - If $\im(\phi) \subseteq K\ts{\tau}$ for some subfield $K\leq L$ we say $\phi$ is **defined over $K$**. - Rank similar to modules, height similar to height of a formal group. - **Ranks**: $\phi$ satisfies ultrametric properties so the map $f(a) = \exp(\deg \phi_a)$ (considering $\phi_a$ as a polynomial in $T$) gives a place which turns out to be equivalent to $\infty$, so $\exists d$ such that $\deg \phi_a = d v_\infty(a) \da d \deg a$, define $\rank \phi \da d$. - **Heights**: the map $g(a) = \exp(-m(a))$ where $m$ is the least integer such that $\tau^m$ appears in $\phi_a$, or $0$ if not. This is equivalent to the place $\mfp$, so $\exists h$ with $m(a) = h v_{\mfp}(a)$, define $\height \phi \da h$. - **Torsion**: let $\bar L$ be an algebraic closure and define $$\phi[a] \da\ts{x\in \bar L \st \phi_a(x) = 0} \qquad \in \mods{A/a}, \qquad \phi[I] \da\Intersect_{a\in I} \phi[a] \text{ for } I\normal A$$ - Can form analogs of Eisensteins series, $\HH$ (the Drinfeld upper half plane), the $j\dash$invariant, and (Drinfeld) modular forms. - Def 4.4.3: A **morphism** of Drinfeld modules $\phi, \psi$ over an $A\dash$field $F$ is a polynomial $P\in F\ts{\tau}$ with $P\phi_a = \psi_a P$ for all $a\in A$. An **isogeny** is a nonzero morphism. ## Example: the Carlitz Module - The Carlitz module: $A = \FF_q[T]$ with $$\begin{align}\iota: A&\to \CC_\infty \\ a &\mapsto \rho_a \end{align}$$ yields a Drinfeld module with generic characteristic. Defined by $\phi_T = \iota(T) + \tau$, so rank 1 (**unique** Drinfeld of rank 1) - Defining the "addtive polynomial" $\rho_a$: - - Properties: - $e_C(x+y)=e_C(x)+e_C(y)$; - $e_C(\alpha x)=\alpha e_C(x)$ for any $\alpha \in \mathbf{F}_q$; - $e_C(T x)=T e_C(x)+e_C(x)^q$. - This implies that for any $a \in A$ there is a polynomial $$\rho_a(x)=a x+c_1 x^q+\cdots+c_d x^{q^d}, \quad d=\operatorname{deg}(a), \quad c_1, \ldots, c_d \in \mathbf{C}_{\infty}, c_d \neq 0,$$such that $$e_C(a \cdot x)=\rho_a\left(e_C(x)\right)$$, the analog of $e^{ax} =(e^x)^a$ classically. - E.g. take $a=T$ to get $p_T(x) = Tx + x^q$, recovering 3 above. - $\exp _C: \CC_{\infty} \rightarrow \CC_{\infty}$ is an $\FF_q[T]$-module homomorphism, where $t$ acts on the domain by scalar multiplication by $T$ and on the range by the endomorphism $C(t)$. - Use $\left(C, \CC_{\infty}\right)$ to denote $\CC_{\infty}$ with the Carlitz $\FF_q[T]$-module structure. We have a "uniformization sequence" $$0 \rightarrow \FF_q[T] \cdot \pi_C \rightarrow \CC_{\infty} \stackrel{\exp _C}{\longrightarrow}\left(C, \CC_{\infty}\right) \rightarrow 0 \qquad\in\mods{\FF_q[T]}$$ - $\pi_C$ is called the **Carlitz period**. - This sequence used in (Wade 41) shows $\pi_C$ is transcendental over $\FF_q(T)$, among first transcendence results in positive characteristic - "Division values" (torsion?) generate abelian cyclotomic extensions of $\FF_q(T)$ - For the Carlitz module, one can construct abelian extensions of $K$ by adjoining torsion $\phi[a]$ for various Drinfeld modules $\phi$: if $N\normal A$ is generated by $\prod (T-t_i)$ with $t_i\in k$ distinct, then $$K(\phi[N]) = K(\sqrt[q-1]{T-t_1}, \cdots, \sqrt[q-1]{T-t_n})$$ which is Galois with group $(A/N)\units$. ## Lattice Exponentials - An $A\dash$submodule $\Lambda \subset \CC_\infty$ is an **$A\dash$lattice** if - $\Lambda \in \amod^\fg$ - $\Lambda$ is discrete in the topology of $C_\infty$ - If $\Lambda \subset M^\sep$ and is stable under $\Gal(M^\sep/M)$, then $\Lambda$ is an $M\dash$lattice. - $\rank \Lambda$ is its rank as a fg projective torsionfree submodule of $\CC_\infty$. - Starting with an A-lattice $\Lambda \subseteq \mathbf{C}_{\infty}$ we define the **lattice exponential function** $$\exp _{\Lambda}(z):=z \prod_{\lambda \in \Lambda\smz}\left(1-\frac{z}{\lambda}\right), \quad z \in \mathbf{C}_{\infty}$$ - Converges by discreteness of $\Lambda$ (finitely many points in any fixed ball about the origin) - Considering partial products yields a $\FF_q\dash$linear power series $$\exp _{\Lambda}(z)=z+\sum_{i \geq 1} a_i z^{q^i},$$ so $$\exp _{\Lambda}\left(z_1+c z_2\right)=\exp \left(z_1\right)+c \exp _{\Lambda}\left(z_2\right),\,\, c\in \FF_q \qquad \exp _{w \Lambda}(w z)=w \exp _{\Lambda}(z),\,\, w\in \CC_\infty$$ - $e_\Lambda(z)$ is $\FF_q\dash$linear and yields an isomorphism $\CC_\infty/\Lambda \isoas{\zmod} \CC_\infty$ - **Prop 4.2.5**: $e_L(x)$ is $\FF_r\dash$linear. - Proof: write $L = \Union_{i\geq 0} L_i$ with $L_i\in \mods{\FF_q}^{\fd}$ to get $e_L(x) = \lim_i e_{L_i}(x)$; each finite stage is $\FF_r\dash$linear, now apply Cor 1.2.2. - **Cor 4.2.6**: $e_L(x)$ yields an isomorphism $C_\infty/L \isoas{\zmod} C_\infty$. - Yields a uniformization exact sequence $$\Lambda \injects \CC_\infty \surjectsvia{\exp_\Lambda} \CC_\infty$$ - For $a\in A$, in general one can write $$\exp _{\Lambda}(a z)=\phi_{\Lambda}(a) \exp _{\Lambda}(z)$$ where $\phi_\Lambda(a): A\to \CC_\infty\ts{\tau}$, so any lattice defines a Drinfeld $A\dash$module $\phi_\Lambda$ ## Main Theorem - **Theorem (Drinfeld's Uniformization Theorem)** - Given a homomorphism $\phi: A \rightarrow$ $\mathbf{C}_{\infty}\{\tau\}$ such that $$\phi(a)=a+\cdots+a_m \tau^m, a_m \neq 0,\qquad m=r \operatorname{deg}(a)$$, there is a unique A-lattice $\Lambda$ such that $\phi=\phi_{\Lambda}$. Moreover $\operatorname{rank}_A \Lambda=r$. - **Main theorem**: Analytic Uniformization - Let $\psi$ be a Drinfeld module over $M$ of rank $d>0$. Then there is an $M\dash$lattice $L(\psi)$ depending on $\psi$ of rank $d$ such that $\psi$ is its associated Drinfeld module; the functor $\psi \mapsto L(\psi)$ induces an equivalence of categories between Drinfeld modules of rank $d$ and $M\dash$lattices of rank $d$. - Involves defining an entire function $e_\psi(\tau) = \sum c_i \tau^i$ with $c_0 = 1$ which satisfies $e_\psi(ax) = \psi_a(e_\psi(x))$. - **Thm/Def 4.6.10**: For $e_\psi(z)$ the exponential associated to the lattice $L_\psi$, the value $e_\psi(a)$ is transcendental over $k$ for all $a\in \kbar\smz \subseteq \CC_\infty$, so all elements of $L_\psi$ are $k\dash$transcendental. The elements of $L_\psi$ are called **periods** of $\psi$. - Motivation: strong similarty between Drinfeld modules of a fixed rank over $C_\infty$ and elliptic curves over $\CC$. - The gist: $\psi$ can be recovered from a lattice and its exponential - Remainder of the chapter: develop a theory that parallels elliptic curves (CM by maximal orders, conductors, stable/good reduction) ## Modular Theory - Define the **Drinfeld symmetric space** $\Omega^r = \PP^{r-1}(\CC_\infty) - \Union_{H\in S} H$ where $S$ consists of all $F_\infty\dash$rational hyperplanes, then $$\ts{\text{Rank $r$ Drinfeld modules over }\CC_\infty}\mapstofrom \dcosetl{\GL_r(A)}{\Omega^r}$$ - This receives the structure of a nonarchimedean space using the Bruhat-Tits building of $\PGL_r(\FF_\infty)$. - For $r=2, \Omega^2 = \CC_\infty \sm \RR_\infty$, compare to elliptic curves being classified by $\dcosetl{\GL_2(\ZZ)}{\CC\sm\RR}$. - Can take $\dcosetl{\Gamma}{\Omega^r}$ to take moduli with level structure, analogs of modular curves. - E.g. for $\mfn\in A$ a nonzero polynomial, $$\Gamma_0(\mathfrak{n}):=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in \mathrm{GL}_2(A) \mid c \equiv 0(\bmod \mathfrak{n})\right\}\leadsto Y(\mfn) \da\dcosetl{\Gamma(\mfn)}{\Omega^2}$$ an affine curve over $F$, contained in a unique smooth projective curve $X(\mfn)$. - For rank 2 Drinfeld modules: write $\phi_T = i(T) + g\tau + \Delta\tau^2$ and define $j(\phi) \da g^{q+1}/\Delta$ to get an analog of the **$j\dash$invariant**; this classifies such rank 2 Drinfeld modules up to iso. ## Shtukas - Defining **shtukas**: - $X\slice k$ a curve over a finite field - For $S\in\Sch\slice k$, write $X_S \da X\fiberprod{k} S$ for the base change - Write $\Frob_S \da \id_X \fiberprod{k} \Frob_S$ as a morphism $X_S\to X_S$. - For $X\in \Sch\slice A$ with structure morphism $i: X\to \spec A$, a Drinfeld $A\dash$shtuka over $S$ is a cospan: ![](attachments/2023-01-12-5.png) where $F, F' \in \mods{\OO_X}^{\loc\free}$ are rank $d$, with conditions on the cokernels. - The composite $\beta\circ\alpha\inv$ defines a rational map $\phi: \Frob^*_S F\torational F$ which which is regular away from $\infty$ in $S$. - Why care: let $\infty \in X(k)$ be a $k$-rational point, and let $A=H^0\left(X \backslash\{\infty\}, \mathcal{O}_X\right)$. - Let $S \rightarrow \operatorname{Spec} A$ be a scheme. The following categories are equivalent: A. Drinfeld A-modules of rank $d$ over $S$. B. Drinfeld A-shtukas of rank d over $S$ with $\chi(\mathcal{F})=0$.