--- date: 2023-01-19 18:00 aliases: ["Notes Talk 2"] --- Last modified: `=this.file.mday` # Notes Talk 2 ## Opening Remarks - **Side board** - $K$ the function field of a curve $X$, global of finite characteristic. - E.g. $K = \FF_q(T)$. - $K_\infty$ its (metric) completion at a fixed place $\infty$ $(\simeq \RR)$. - $\CC_\infty = \widehat{ \bar{K} }$: extend $v_\infty$ to $\bar{K}$ and complete. - $A\subseteq H^0(X\smts{\infty}; \OO)$ a subalgebra of functions regular away $\infty$. - Replace $\ZZ$ by $A \da \FF_q[T]$ and $\zmod$ by $\amod$. - E.g. $A = \FF_q[T]$. - $L$ an $A\dash$field, i.e. a pair $(L,\iota\in \algs{\FF_q}( A\to L))$ - Note $\spec L \to \spec A$, so so $L\in \algs{\FF_r}\slice{A}$, - $\mfm \da \ker \iota$ the characteristic of $L$ - $L\ts{\tau} \subseteq \Endo_L(\GG_a)$ a twisted noncommutative polynomial subalgebra - Idea: generalize number fields to global fields, get an arithmetic theory in characteristic $p$ by working with function fields of curves. - The Carlitz module: $a\mapsto C_a(u)$ where $e_C(a x) = C_a(e_C(x))$ for $C_a$ an additive polynomial - Leads to a good theory of nonarchimedean complex analysis for a rational function fields, with analogues of factorials $n!$, Bernoulli numbers $B_k$, zeta functions $\zeta_C(x)$, "division values" (torsion points) of $e_C$ generate good "cyclotomic" extensions of the base field. - Drinfeld: generalizes to "Elliptic modules", higher rank exponential, extend to arbitrary function fields, more algebraic objects, moduli of Drinfeld modules uniformized using Tate's rigid analytic spaces. - Idea: generalize lattice theory (free $\zmod$) and orders to other global fields to get a theory of CM for function fields - Drinfeld modules $\simeq$ elliptic curves, - $T\dash$modules $\simeq$ abelian varieties (higher dimension Drinfeld modules) - $T\dash$motives $\simeq$ ??? ## Explicit class field theory - Fields $L\in \mods{A}$ gain a new $A\dash$module structure by $\psi$ a Drinfeld module. - Explicit class field theory via the Carlitz module - Replace $\ZZ$ by $A \da \FF_q[T]$ and $\zmod$ by $\amod$. - $\ZZ\simeq \FF_q[T]$ and $\ZZ\units \simeq \FF_q[T]\units = \FF_q\units$. - $m\in \ZZ\smz, M\in \FF_q[T]\smz \leadsto \ZZ/m, \FF_q[T]/M$ finite rings. - $\ZZ/m = \Gal(\QQ(\mu_m)/\QQ)$ and $(\FF_q[T]/M)\units = \Gal(K/\FF_q(T))$ for some "cyclotomic" $K$. - Cyclotomic polynomials $x^m-1$ for $m\in \ZZ$, roots form a $\ZZ\dash$module, adjoin roots to get $\QQ(\mu_m)$ (and thus abelian extensions by Kronecker-Weber); - analogous $[M](x)$ parameterized by $M\in \FF_q[T]$, roots form an $A\dash$module, adjoining root to $\FF_q(T)$ to get $K$ (and thus abelian extensions: Carlitz-Hayes theorem - For $M\in \FF_q[T]$ define $[M]$ by forcing $\FF_q\dash$linearity: $$[M](x) = \sum_{i=1}^n c_n[T^n](x) \in\FF_q[T][x]$$ ## Defining Drinfeld modules - Standard definition; - Define a Drinfeld $A\dash$module $\rho$ over $L$ as $\rho\in \algs{\FF_r}(A\to L\ts{\tau})$ commuting with $D$ which is nontrivial in the sense that $\rho \neq \eps\circ \iota$ where $\eps: L\injects L\ts{\tau}$ is the inclusion. - A **Drinfeld module of rank $r$ over $L$** is a morphism $$\psi: A\to L\ts{\tau} \qquad\in \algs{\FF_q}, \qquad \psi(T) = \iota(T) + \bigo(\tau)$$ - ![](attachments/2023-01-12diag.png) - Here $r$ is the highest power of $\tau$ that appears in $\bigo(\tau)$. - Geometrically, $\psi$ is $\GG_a/L$, but think of it as a pair $(\psi, \GG_a)$ where $\psi: A\to \Endo_L(\GG_a)$. - 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 - Alternative definition of Drinfeld module: - $K$ a local field with valuation ring $\OO_K$ - $L\in \algs{\OO_K}$ with structure map $\gamma: \OO_K\to L$, - Define a **formal Drinfeld $\OO_K\dash$module** (over $L$ ) as a pair $(F, f)$ where - $F\in L\fps{x, y}$ is a formal group and - $f: \OO_K \to \Endo_L(F)$ commuting with the derivative $D$, so $D\circ f = \gamma$. - Where $D: \Endo_L(F)\to L$ where $g\mapsto g'(0)$, - For $F = \GG_a$, so $F(x,y) = x + y$, identify $\Endo_L(\GG_a) \cong L\ts{\ts{\tau}}$ the twisted (formal) power series ring and $D$ sends a series to its constant term. - $\psi$ has **generic characteristic** if $\iota: A\to L$ injects, otherwise $\psi$ has **characteristic $\mfp$** where $\mfp \da \ker \iota$ - If $\im(\psi) \subseteq K\ts{\tau}$ for some subfield $K\leq L$ we say $\psi$ is **defined over $K$**. - The map $g(a) = \exp(-m(a))$ where $m$ is the least integer such that $\tau^m$ appears in $\psi_a$, or $0$ if not. - This is equivalent to the place $\mfp$, so $\exists h$ with $m(a) = h v_{\mfp}(a)$, define $\height \psi \da h$. Call $h$ the **height**. - $\psi$ satisfies ultrametric properties so the map $f(a) = \exp(\deg \psi_a)$ (considering $\psi_a$ as a polynomial in $T$) gives a place equivalent to $\infty$, - So $\exists d$ such that $\deg \psi_a = d v_\infty(a) \da d \deg a$, define $\rank \psi \da d$ to be the **rank**, - **Torsion**: let $\bar L$ be an algebraic closure and define $$\psi[a] \da\ts{x\in \bar L \st \psi_a(x) = 0} \qquad \in \mods{A/a}, \qquad \psi[I] \da\Intersect_{a\in I} \psi[a] \text{ for } I\normal A$$ - A **morphism** of Drinfeld modules $\psi, \psi$ over an $A\dash$field $L$ is a polynomial $P\in L\ts{\tau}$ with $P\psi_a = \psi_a P$ for all $a\in A$. - An **isogeny** is a nonzero morphism. - CM: - $E$ a finite extension of $K \subseteq \CC_\infty$ is a **$\mathrm{CM}_\infty\dash$field** if $E$ contains exactly one prime above $\infty$. - A Drinfeld module $\psi$ **has CM by the maximal order $\OO$ of $\AA\dash$integers** in $E$ if $\OO\injects \Endo(\psi)$. - Let $\Lambda$ be a lattice for $\OO$ of rank $d_1$, then $\Lambda$ is an $A\dash$lattice of rank $d\da d_1[E: k]$, so say **$E$ has CM** iff $d=1$ iff $L\cong I$ where $I$ is an $\OO\dash$ideal. ## Uniformization ### Motivation and analogies - $\GG_m$ is uniformizable via $$2\pi i \ZZ \injects \CC\surjectsvia{\exp} \CC\units$$ - Say $\alpha\in \CC$ is **torsion** for $\exp$ if $\alpha\in \ker \exp$, so $\exp(\alpha) =0$. - Torsion values are of the form $\zeta_n$, note these generate cyclotomic extensions of $\QQ$. - Can ask transcendence questions about periods, exponentials, and logs of algebraic numbers. - **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 $\QQ(\zeta_m)/\QQ$? - Note that $\CC\units = \GL_1(\CC)$ and $\Lie(\GL_n(\CC)) = \Mat_{n\times n}(\CC)$, so we can rewrite this as $$2\pi i \ZZ \injects \Lie(G) \surjects G \qquad G = \CC\units = \GL_1(\CC)$$ - Define an "exponential" for elliptic curves $\exp_E(z) \da \tv{\wp(z), \wp'(z), 1}$ to get $$\Lambda \injects \CC \surjectsvia{\exp_E} E(\CC)$$ where $\Lambda$ is the **period lattice** of $E$, $\Lambda = \gens{1, \tau}_\ZZ$. - When $E$ is defined over $K\in \NF$, torsion values generate abelian extensions of $K$. - Recall $$\wp(z, \Lambda):=\frac{1}{z^2}+\sum_{\lambda \in \Lambda \backslash\{0\}}\left(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2}\right)$$ which is designed to have an order two pole at every point in $\Lambda\smz$. - Can ask transcendence questions about periods and elliptic logs of algebraic points - For any abelian variety $A$, $A(\CC)$ forms a compact complex Lie group! So here we have $$\Lambda \injects \Lie(G) \surjectsvia{\exp_G} G \qquad G = E(\CC)$$ - For $G$ a commutative algebraic group, one generally has $$\Lambda \injects \Lie(G) \surjectsvia{\exp_G} G(\CC)$$ where $\Lambda$ is a lattice in the Lie algebra of $G$. - One can study special values of $\exp_G$ and logs of algebraic points on $G$. - Recall $\Lie(G) = \T_0 G$, and it turns out that the torsion $G[n]$ is isomorphic to $n\inv \Lambda/\Lambda \cong \Lambda/n\Lambda$. - For the Carlitz module: - Let $A = \FF_q[T]$, $\CC_\infty = ?$. - $\left(C, \CC_{\infty}\right)$ denotes $\CC_{\infty}$ with the "exotic" Carlitz $\amod$ structure, regard as a rank 1 Drinfeld module $\psi$. - We have a uniformization sequence$$\Lambda \injects (\triv, \CC_{\infty}) \surjectsvia{e_C}\left(C, \CC_{\infty}\right) \qquad\in\mods{\FF_q[T]}, \,\quad\Lambda = \gens{\pi_C}_A$$ - $\pi_C$ is called the **Carlitz period**, denoted $\xi$ in the book. - Wade (41) shows it is transcendental over $K\da \ff(A) = \FF_q(T)$, torsion values $\psi[a]$ generate abelian "cyclotomic" extensions of $K$. ### 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 $\CC_\infty$ (equiv: any finite ball in $\CC_\infty$ contains finitely many elements of $L$) - $\rank \Lambda$ is its rank as a fg projective torsionfree submodule of $\CC_\infty$. - Let $M$ be a complete extension of $K\subseteq \CC_\infty$. If $\Lambda \subset M^\sep$ and is stable under $\Gal(M^\sep/M)$, then $\Lambda$ is an **$M\dash$lattice**. - Its **rank** is its rank as a fg torsionfree sub $A\dash$module 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 $$\exp _{\Lambda}(a z)=\psi_{\Lambda}(a) \exp _{\Lambda}(z)$$ where $\psi_\Lambda(a): A\to \CC_\infty\ts{\tau}$, so any lattice defines a Drinfeld $A\dash$module $\psi_\Lambda$ ### Uniformization theorem - **Theorem (Drinfeld's Uniformization Theorem)** - Given a homomorphism $\psi: A \rightarrow$ $\mathbf{C}_{\infty}\{\tau\}$ such that $$\psi(a)=a+\cdots+a_m \tau^m, a_m \neq 0,\qquad m=r \operatorname{deg}(a)$$, there is a unique A-lattice $\Lambda$ such that $\psi=\psi_{\Lambda}$. 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 $$\ts{\text{Drinfeld modules of rank } r}\mapstofrom\ts{M\dash\text{lattices of rank }r }$$ - 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 $\CC$. - **TLDR**: $\psi$ 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 # Closing comments - Define a **Drinfeld symmetric space** $\Omega^r \subseteq \PP^{r-1}(\CC_\infty)$ - This receives the structure of a nonarchimedean space using the Bruhat-Tits building of $\PGL_r(\FF_\infty)$. - Theorem: $$\ts{\text{Rank $r$ Drinfeld modules over }\CC_\infty}\mapstofrom \dcosetl{\GL_r(A)}{\Omega^r}$$ - For rank 2 Drinfeld modules: write $\psi(T) = i(T) + g\tau + \bigo(\tau^2)$ and define $j(\psi) \da g^{q+1}/\Delta$ to get an analog of the **$j\dash$invariant**; this classifies such rank 2 Drinfeld modules up to iso. - Theorem: let $S \rightarrow \operatorname{Spec} A$ be a scheme. The following categories are equivalent: - Drinfeld $A\dash$modules of rank $r$ over $S$. - Drinfeld $A\dash$**shtukas** $\mcf$ of rank $r$ over $S$ with $\chi(\mathcal{F})=0$. - Set $F = \tau^s$ where $q=p^s$ and $V \da V_w(\phi) \da T_w(\phi)\tensor K$ where $w$ is a prime of $A$ not equal to the characteristic of $\psi$, then there is a zeta function $$\sum_{k\geq 1}\det\qty{1- F^k \mid V}u^k = \dd{}{u}\log Z(u), \quad Z(u) \da \prod_{0\leq i\leq d} f_i(u)^{(-1)^{i+1} }, \qquad f_i(u) = \det\qty{1-uF \mid \wedge^i V}$$ These determine a Drinfeld module up to isogeny (along with the rank), since these modules biject with Weil numbers (similar to reciprocal roots in Weil conjectures)