--- date: 2023-01-11 19:44 aliases: ["Reading group winter 2022 talk notes 2"] --- Last modified: `=this.file.mday` --- - Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links --- # Reading group winter 2022 talk notes 2 ## Carlitz Modules - Symbols: ![](attachments/2023-01-11symbols.png) - The Carlitz module $C$ is the first example of a Drinfeld module. Defined by Carlitz [9] in 1935 , it is given by the $\FF_{q^{-} \text {algebra homomorphism }}$ $$ C: \FF_q[T] \rightarrow \CC_{\infty}\{\tau\} $$ defined so that $$ C(t)=T+\tau . $$ - A twisted polynomial $f=a_0+a_1 \tau+\cdots+a_d \tau^d \in$ $\CC_{\infty}\{\tau\}$ represents the $\FF_q$-linear endomorphism of $\CC_{\infty}$, $$ x \mapsto f(x)=a_0 x+a_1 x^q+\cdots+a_d x^{q^d} . $$ - $C$ makes $\CC_{\infty}$ into an $\FF_q[T]$-module, where $$ a.x \da C(a) x,\qquad C(t)(x)=T x+x^q . $$ Exponential functions enter the picture with the Carlitz exponential function $$ \exp _C(z)\da\sum_{i \geq 0} \frac{z^{q^i}}{D_i}, $$ where $D_0=1$ and - $$D_i=\left(T^{q^i}-T\right)\left(T^{q^i}-T^q\right) \cdots\left(T^{q^i}-T^{q^{i-1}}\right),\qquad i\geq 1$$ for $i \geq 1$. - This function converges for all $z \in \CC_{\infty}$, and the recursion $D_i=\left(T^{q^i}-T\right) D_{i-1}^q$ implies that $$ \exp _C(T z) = C(t)\cdot \exp _C(z) $$ - Carlitz's motivations: - Explicit class field theory for the rational function field $\FF_q(T)$. - For $f \in \FF_q[T]$, define the $f\dash$torsion on $C$ as $$C[f]:=\left\{x \in \CC_{\infty} \mid C(f)(x)=0\right\} \isoas{\mods{\FF_q[T]}} \FF_q[T]/(f)$$which is a Galois module over the separable closure of $\FF_q(T)$. - The Carlitz cyclotomic field is the field $\FF_q(T, C[f])$, and there is an isomorphism $$ \rho:\left(\FF_q[T] /(f)\right)^{\times} \stackrel{\sim}{\longrightarrow} \operatorname{Gal}\left(\FF_q(T, C[f]) / \FF_q(T)\right), $$ such that for an $\FF_q[T]$-module generator $\lambda \in C[f]$ we have $$\rho_a(\lambda)=C(a)(\lambda)\qquad a \in\left(\FF_q[T] /(f)\right)^{\times}$$ - $\rho_a$ coincides with the Artin automorphism for $a$, - In this way we obtain an explicit Galois action on a piece of the maximal abelian extension of $\FF_q(T)$ that agrees with class field theory. - Hayes later completes the picture to get an analog of the Kronecker-Weber theorem: gets a Galois action on the maximal abelian extension of $\FF_q(T)$ ## Drinfeld Modules - Analogs with elliptic curves: ![](attachments/2023-01-11-7.png) - **$t$-motives**. Let $L$ be an extension of $\mathbb{F}_q$, let $\iota: \mathbb{F}_q[T] \rightarrow L$ be an $\mathbb{F}_q$-algebra homomorphism, and set $T=\iota(t)$. Let $L[t, \tau]:=L\{\tau\}[T]$ be the ring of polynomials in the commuting variable $t$ over the non-commuting ring $L\{\tau\}$. Thus $$ t c=c t, \quad t \tau=\tau t, \quad \tau c=c^q \tau, \quad c \in L . $$ - A $t$-motive $M$ is a left $L[t, \tau]$-module which is free and finitely generated as an $L\{\tau\}$ module for which there is an $\ell \in \mathbb{N}$ with $$ (t-T)^{\ell}(M / \tau M)=\{0\} . $$ - Morphisms of $t$-motives are morphisms of left $L[t, \tau]$-modules. - The rank $d(M)$ of $M$ as an $L\{\tau\}$-module is called the dimension of $M$. - Every $t\dash$module has a unique $t\dash$motive. - Theorem 4.1 (Equivalence of categories (Anderson [1, Thm. 1])). The above correspondence between abelian t-motives and abelian t-modules over $L$ gives an anti-equivalence of categories. - A $t$-motive $M$ is said to be **abelian** if it is free and finitely generated over $L[T]$. - A $t$-module is called **abelian** if its associated $t$-motive is abelian. - Theorem $4.2$ (Anderson). The t-module associated to an abelian t-motive $M$ over $\mathbb{C}_{\infty}$ is uniformizable if and only if $M$ is rigid analytically trivial. - 4.2. The $t$-motive of a Drinfeld module. Let $\phi$ be a rank $r$ Drinfeld $A$-module defined over $L$ by $$\phi(t)=T \tau^0+a_1 \tau+\cdots+a_r \tau^r .$$ - Let $M(\phi):=L\{\tau\}$, and as in the previous section we make $M(\phi)$ into the $t$-motive associated to $\phi$ by setting $$c t^i \cdot m:=c m \phi\left(t^i\right), \quad c \in L, m \in L\{\tau\} .$$ - $M(\phi)$ is an abelian $t$-motive, pure of dimension 1, rank $r$, and weight $1 / r$. ## Misc Other Notes - Drinfeld extended Drinfeld modules to shtukas, and Laurent Lafforgue uses moduli of shtukas to prove Langlands for $\GL_n(K)$ for $K$ a function field. Wide open for $K = \QQ$, even for $n=2$. - $\Lambda\to E_\Lambda$ induces an equivalence between lattices and homotheties $\Hom(\Lambda_1, \Lambda_2) = \ts{\alpha\in \CC\st \alpha \Lambda_1 \subseteq \Lambda_2}$ and elliptic curves. - For $K$ imaginary quadratic, restricts to an equivalence between $\OO_K\dash$stable lattices, i.e. $\OO_K \Lambda = \Lambda$, and elliptic curves $E$ with CM by $\OO_K$. - $E$ can be defined over a finite extension of $K$ in this case: $H\da K(j(E))$. - The analogue of Kronecker-Weber: every abelian extension of $K$ is contained in $H(E[n])$ for some $n$. - Defining function fields: $K/\FF_p(T)$ a finite extension; $\spec K$ is the generic point of a smooth projective integral scheme of dimension 1 over $\FF_p$. - If $v\in \abs{X}$ is a nonarchimedean place, corresponds to a discrete valuation $v:K\to \ZZ\union\ts\infty$. - Set $q\da \size \kappa(v)$ to be the size of the residue field, then $$\abs{f}_v = q^{-v(f)}$$ - For $K = \FF_q(T)$ and $v=\infty$, this becomes $$\abs{f(x)}_\infty = q^{\deg f}, \qquad f\in \FF_q[T]$$ - Satisfies $$\prod_{v\in \abs{K}} \abs{f}_v = 1\quad\forall f\in K\units$$ - The completion $K_v$ for $v$ nonarchimedean is an extension of either $\QQpadic$ or $\FF_p((t))$, and if $K$ is characteristic $p$ then $K_v\cong \FF_q((T))$ where $\FF_q = \kappa(v)$. - Fix $S = \ts{\infty}$, consider $A = \OO_{K, \ts\infty} =H^0(X\sm\infty; \OO_X)$ and set $\CC_\infty \da \widehat{\bar K_\infty}$. - Look at quotients $\CC/\Lambda$ by a discrete $A\dash$submodule (note these quotients are again $A\dash$modules) - Theorem 2.2.1 (Drinfeld). For $z \in C$, let $$\wp_{\Lambda}(z)=z \prod_{\lambda \in \Lambda \backslash\{0\}}\left(1-\frac{z}{\lambda}\right)$$. Then 1. The product defining $\wp_{\Lambda}(z)$ converges in $\CC_\infty$. 2. $\wp_{\Lambda}: \CC_\infty \rightarrow \CC_\infty$ is a surjective continuous homomorphism of topological $\mathbb{F}_{q^{-}}$ vectors spaces, whose kernel is exactly $\Lambda$. Thus $$\CC_\infty / \Lambda \isovia{\wp_\Lambda} \CC_\infty \qquad \in \Top\mods{\FF_q}$$ - Compare to complex analysis: any entire $f$ has a Weierstrass product $p$ with $f=pg$ and $g$ an entire nonvanishing function. Here we don't need the $g$, since the nonarchimedean big Picard theorem applies: there are no nonconstant nonvanishing entire functions on $\CC_\infty$. - Since the LHS $\in\amod$, so is the RHS, but $\wp_\Lambda$ gives $\CC_\infty$ an exotic $\amod$ structure (compared to the one induced by $i: A\to \CC_\infty$). Call $(\CC_\infty, \wp_\Lambda)$ **the Drinfeld module associated to $\Lambda$.** - The underlying $\mods{\FF_q}$ is identical since $\wp_\Lambda$ is $\FF_q\dash$linear; the new thing is $\phi_a\in \Endo(\CC_\infty)$ corresponding to $x\mapsto ax$ on $\CC_\infty/\Lambda$. - Thm: for every $a\in A$, there exists a polynomial $\phi_a\in \CC_\infty[x]$ such that: ![](attachments/2023-01-11diag.png) It is $\FF_q\dash$linear and satisfies $$\phi_\alpha(z)=a_0+a_1 z^q+a_2 z^{q^2}+\ldots,\quad a_i \in \CC_\infty, \qquad\phi_{ab} = \phi_a \circ \phi_b$$ - This defines a morphism $$\begin{align}A &\to \Endo_{\FF_q} \GG_a(\CC_\infty) \qquad \in \algs{\FF_q}\\ a &\mapsto \phi_a \end{align}$$ where $\GG_a(\CC_\infty) = \spec \CC_\infty[z]$ is the additive $\FF_q\dash$module (group) scheme of $\CC_\infty$. - Can recast as a ring morphism $A\to \CC_\infty\ts{\tau}$ where the RHS is a the twisted polynomial ring: a noncommmutative $\CC_\infty\dash$algebra whose underlying module is $\CC[\tau]$ and multiplication is twisted: $\tau a = a^q \tau$. ## The Carlitz Module (again) - Setup: $$\begin{aligned} K & =\mathbb{F}_q(T) \\ A & =\mathbb{F}_q[T] \\ K_{\infty} & =\mathbb{F}_q((1 / T)) \\ \Lambda & =A . \end{aligned}$$ - $\infty \in \PP^1\slice{\FF_q}$ is the point at infinity - $f\in A\implies \abs{f}_\infty = q^{\deg f}$. - $\wp_A: \CC_\infty/ \Lambda \to \CC_\infty$ where $z\mapsto \prod_{a\in A\smz}\qty{1 - {z\over a}}$. - Can define the Carlitz exponential via a product: $e_{\CC_\infty}(z) = \wp_{\xi A}(z)$ for $\xi$ a lattice scaling factor. - Can check $$e_{\CC_\infty}(z) = \sum_{n\geq 0} {z^{q^n} \over D_n} \in K\fps{z}$$ where $D_n$ is like a factorial: $$D_0 = 1,\qquad D_n = (T^{q^n}-T)D_{n-1}^q,\qquad D_n\equalsbecause{\text{thm}}\prod_{\substack{f\in \FF_q[T] \text{monic} \\\deg f = n}} f(T)$$ - There is an expansion $$e_{\CC_\infty}(z) = z \prod_{a\in \FF_q[T]\smz}\qty{1 - {z\over \xi a}}$$ - Compare to classical identity used to prove that $\zeta(2k)\in \pi^{2k}\QQ$ for $k\in \ZZ$: $$\sin (z)=z \prod_{n \geq 1}\left(1-\frac{z^2}{\pi^2 n^2}\right)$$ - Motivates that $\xi \simeq 2\pi i$ - Definition: A **Drinfeld A-module** $\phi$ over $L$ is a homomorphism of $\mathbb{F}_{q^{-}}$ algebras $$\begin{align}A &\rightarrow L\{\tau\} \\ \alpha &\mapsto \phi_\alpha\end{align}$$, such that for all $\alpha \in A$, the constant term of $\phi_\alpha$ is $\iota(\alpha)$ where $\phi$ is not $\alpha \mapsto \iota(\alpha)$. - A morphism $\phi \rightarrow \phi^{\prime}$ between Drinfeld modules is an element $f \in L\{\tau\}$ such that for all $\alpha \in A, f \phi_\alpha=\phi_\alpha^{\prime} f$. - Here $L$ is an $A\dash$field: a field $L$ with a morphism $i: A\to L$ and $\mfp \da \ker i$ is its characteristic. - Equivalently: it is an $A$-module scheme over $L$, whose underlying $\mathbb{F}_q$-module scheme is $\mathbb{G}_{a}(L)$, such that the derivative of $A \rightarrow \Endo_{\mathbb{F}_q} \mathbb{G}_{a, L}$ at the origin is $\iota: A \rightarrow L$. - Drinfeld modules form a category enriched over $A\dash$modules (all homs are). # Motivations - 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: $$\operatorname{Im}\left(J_{4 k-1}\right) \simeq \mathbb{Z} / D_{2 k}\qquad D_{2 k}=\text { the denominator of } B_{2 k} / 4 k \text {. }$$ - 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). - Classical - 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 $$e^z=\sum_{n=0}^{\infty} \frac{z^n}{n !}, \qquad \sinh(z) = \frac{e^z-e^{-z}}{2}=\pi z \prod_{n=1}^{\infty}\left(1+\frac{z^2}{n^2}\right)$$ - Why care: - Relatively easy proof that $\pi_C$ is transcendental over $F$ (Wade, 40s) - Explicit formula for $\zeta_C(k)$ for arbitrary $k$ (Anderson-Thakur, 90s) - Proof that $\zeta_C(k)$ is transcendental over $F$ for all $k$ (Yu, 90s) - Transcendence of $\zeta(n)$ for odd values is wildly open! - Can formula RH for these zeta functions: - Classical: $\zeta(z)$ has a meromorphic continuation to $\CC$ with a simple pole at $z=1$, $\zeta(-k) \in \QQ$ for $k\in \ZZ_{>0}$ and $\zeta(-k) = 0$ for $k\equiv 0\mod 2$, and conjecturally only has zeros on the line ${1\over 2} + it$. - Goss (79) extends $\zeta_C$ to a Carlitz-Goss zeta function from $\ZZ_{> 0}$ to $\CC_\infty\units\times \ZZpadic$ and proves $\zeta_C(-m)\in A$ for $m \in \ZZ_{>0}$ and $\zeta(-m) = 0$ when $k\equiv 0 \mod (q-1)$, - **Moreover** there is a proof that all zeros of $\zeta_{\mathrm{CG}}(x,-y)$ are simple and all lie on the "line" $F_\infty \da \FF_q\fls{T\inv}$ (the completion of $\FF_q(T)$ with respect to its $T\dash$adic norm $\norm{p\over q} = q^{\deg p - \deg q}$). - Proof due to Wan 96, Thakur-Diaz-Vargas 96, Sheats 98 - 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... **- Notes about previous talks:** - Carlitz exponential defined by Peter $e_d(x)$ (product and sum forms), addition property, definition of $D_n$, product of monics, $A\dash$module structure due to $\CC_\infty/\Lambda\iso \CC_\infty$, generating abelian extensions, characteristic of $A\dash$fields - By Haiyang: additive polynomials, $k\ts{\tau}$, $\FF_q\dash$linearity, left division algorithm, $\Endo_{\FF_q}(\FF_q^n) \subseteq \FF_{q^n}\ts{\tau}$.