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date: 2023-01-11 00:08
title: Basic Structures of Function Field Arithmetic Goss
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- Basic Structures of Function Field Arithmetic Goss
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# Basic Structures of Function Field Arithmetic Goss
## Ch. 1
- Essential feature of Wiles FLT: arithmetic cohomology.
- Put a topology on the algebra of functions, vs the discete set of solutions over $\FF_q$.
- RH for curves: uses the Jacobian.
- Idea (Carlitz): instead of attaching a zeta function $\zeta_C(z)$, attach an exponential $e_C(z)$: similar to $e^z$, but domain and codomain are characteristic $p$.
- $e_C(\ell z) = C_\ell(e_C(z))$ where $C_\ell$ is an additive polynomial: analog of $e^{\ell z} = (e^z)^\ell$.
- The map $\ell \mapsto C_\ell(z)$ is the Carlitz module.
- Yields a theory of $n!$ (i.e. $\Gamma(n)$) and Bernoulli functions.
- "Division values" of $e_C(z)$ generate abelian cyclotomic extensions of $\QQ$, analogous to division values of $e^{2\pi i z}$.
- Has "rank" 1.
- Drinfeld generalizes in 74: general exponentials of higher rank, for arbitrary function fields and choices of infinite places.
- The Carlitz module is the simplest Drinfeld modules.
- Constructs a moduli of such modules
- Uiformize the moduli using Tate's rigid analytic geometry.
- His goal: prove Langlands-esque reciprocity laws.
- Yields class fields.
- Generalize elliptic curves and higher so-called $T\dash$modules which generalize abelian varieties.
- Generlize *to* shtukas: Drinfeld modules with a vector bundle.
- Hayes develops an alternative theory, more elementary, uses this for explicit class field theory.
- Toward analogs of $L\dash$functions for abelian varieties and analogs of RH for them: fix $K$ a function field,
- $A \da$ functions in $K$ which are regular away from $\infty$, generalizes $\FF_q[T]$, is a Dedekind domain with finite unit group and class number.
- Given a Drinfeld module $\psi$ over a field $L$, this gives $L$ a new $A\dash$module structure. Use this to define an analog of the Tate module of $\psi$.
- The charpoly of $\Frob_L$ has coefficients in $A$ (instead of $\ZZ$), assemble these into an Euler product and thus an $L\dash$function.
- No analogous function equation like $s\mapstofrom 1-s$.
- Three themes:
- Function fields are "bigger" than number fields: over $\QQ$, analytically complete and then take a degree 2 field extension to close algebraically to get $\CC$, but for function fields, completing yields a local field which may have infinitely many extensions of a fixed degree. Things identified in the number field case split up in the function field case.
- Arithmetic/geometric dichotomy: function fields have two types of cylcotomic extensions, adjoining $\zeta_n$ (everywhere unramified, related to extensions of constant field) or ramified "geometric" extensions (by division values of Drinfeld modules).
- Two $T$'s, as an operator (gives rise to $L\dash$functions and a scalar (gives rise to $\Gamma\dash$functions).
## Ch. 2
## Ch. 3
## Ch. 4
### 4.1 Intro
- Main fetures of Drinfeld modules:
- The lattice $\FF_q[T]\xi$
- Its exponential, similar to $e_C(x)$
- The multiplication formula for $e_C(x)$, similar to the Carlitz module $C$
- Algebraicty of $C$ and reduction to "finite fields"
- The action of $\FF_q[t]$ on rational points via $C$.
- Notation:
- $X$ smooth projective geometrically connected curve over $\FF_r$ where $r = p^{m_0}$.
- $\infty\in X$ a fixed closed point of degree $d_\infty$ over $\FF_q$.
- $k$ is the function field of $X$
- $A \subset k$ the functions regular away from $\infty$
- $v_\infty$ the valuation at $\infty$
- $K \da k_\infty$ its completion at $\infty$.
- $C_\infty$ the completion of $\algcl K$ (algebraically closed)
- $\deg D$ for $D\in \Div(X)$ is the degree of $D$ over $\FF_r$.
- $d_\infty = \deg \infty$.
- For $x\in K$, $\deg(x) \da -d_\infty v_\infty(x)$ so $\deg(a) = \log_r(\size \AA/a)$ for $a\in A\smz$ (the degree of the finite part of the divisor of $a$)
- $\abs{x}_\infty \da r^{\deg x}$the normalized absolute value, extended to $C_\infty$.
- $Jac_X(\FF_q)$ the (finite) group of degree zero divisors on $X$, mod divisors of elements of $k\units$.
- $h(k) = \size \Jac_X(\FF_q)$ the "class number" of $k$. $h(A)$ is the class number of $A$ as a Dedekind domain.
- Facts:
- $A$ is Dedekind at $\spec A = X\smts{\infty}$
- $A\units = \FF_q\units$ since every nonconstant $a\in A$ needs zeros to balance out the pole at $\infty$ since $\deg a = 0$ for any principal divisor $a$.
- Prop 4.1.1: for $I \subseteq k$ a nontrivial $A\dash$fractonal ideal, $I\subset K$ is discrete and cocompact.
- Discreteness: $\exists a\in k\units$ st $aI \subseteq A$.
- Compactness of $K/I$: Riemann-Roch
- Lem 4.12: there is a SES $\Jac_X(\FF_q) \injects \Cl(A) \surjects \ZZ/(d_\infty)$ where the middle is the usual class group of a Dedekind domain.
- Cor 4.13: $h(A) = d_\infty h(k)$.
- Analogies:
- $\ZZ\leadsto A$
- $\QQ\leadsto k$
- $\RR\leadsto K$
- Motto: function field theory decomposes classifical number-theoretic objects into arithmetic components.
- Example: will need many Hilber class fields to play the role of $\QQ$,
### 4.2 Exponentials associated to lattices
- Definition 4.2.1 ($M\dash$lattices)
- 
- Def 4.2.3: for $L$ as above, define $e_L(x) \da x\prod_{a\in L\smz} \qty{1 - {x\over a}}$.
- Prop 4.2.4: $e_L(x)$ is entire on $C_\infty$ which has a Taylor expansion at $x=0$ with coefficiets in $M$.
- Proof: convergence is due to discreteness of $L, the Taylor expansion is from $G_M \da \Gal(M^\sep/M$) which fixes $L$ and the natura of the continuous action $G_M\actson M^\sep$.
d it $e_L$ is a non-constant entire function and thus surjective.
### 4.3 Drinfeld modules of a lattice
- Notation: $L$ a lattice associated to an extension $M/K$ inside $C_\infty$, $d\da \rank_A(L)$.
- Thm 4.3.1 (fundamental) $$e_L(ax) = ae_L(x)\cdot \prod_{a\in (a\inv L/L)\smz} \qty{1 - {e_L(x)\over e_L(a)}}$$
- Proof: short.
- Prop 4.3.2: define $$\begin{align} A &\to M\ts{\tau} \\ a&\mapsto \phi_a\end{align}$$ satisfies:
- $\FF_r\dash$linear,
- $a\in \FF_r \subset A\implies \phi_a = a \tau^0$,
- $\phi_{ab}(\tau) = \phi_a(\tau)\phi_b(\tau) = \phi_b(\tau)\phi_a(\tau) = \phi_{ba}(\tau)$.
- Definition 4.3.3: the Drinfeld module of $L$ is the injection $$\begin{align} A &\to M\ts{\tau} \\ a&\mapsto \phi_a\end{align}$$ which has rank $d = \rank_A(L)$.
- Def 4.3.4 Morphisms of $A\dash$lattices: for $L_1, L_2$ two $A\dash$lattices of the same rank, a morphism $f: L_1\to L_2$ is an element $c\in C_\infty$ with $cL_1 \subseteq L_2$. If $L_1, L_2$ are $M\dash$lattices for some complete field $K\subseteq M\subseteq C_\infty$, and $M\dash$morphism is a morphism $f:L_1\to L_2$ induced by some $c\in M \subseteq C_\infty$.
- Prop 4.3.5: for $\phi, \psi$ two Drinfeld modules with $L_1, L_2$, a morphism $c: L_1\to L_2$ corresponds to a polynomial $P_c(\tau)\in C_\infty(\tau)$ with $P\phi_a = \psi_a P$ commuting with $e_{L_1}$ and $e_{L_2}$: 
## 4.4 General definition of a Drinfeld module
- Def 4.4.1: an $A\dash$field $F$ is a field with a morphim $i:A\to F$; $\ker i =\mfp$ is prime and $\characteristic F \da \mfp$; $F$ has "generic characteristic" (char zero) if $\mfp = \gens{0}$, otherwise $F$ has finite characteristic.
- Agrees with definition for $A = \FF_r[T]$.
- Define $F\ts{\tau}/F$ a ring where $\tau$ is the $r$th power map.
- Def 4.4.2 (Fundamental) $\phi\in \Hom_{\algs{\FF_r}}(A, F\ts{\tau})$ is a Drinfeld module over $F$ iff $D\circ \phi = i$ and $\phi_a \neq i(a)\tau^0$ for some $a\in A$, where $D$ is defined by writing $f(\tau) = \sum_{i=0}^n a_i \tau^i\in F\ts{\tau}$ and setting $Df\da a_0 = f'(\tau)$.
- Example: for $A = \FF_r[T], k = \FF_r(T)$, let $F = k$ and $C: A\to k\ts{\tau}$ with $C_T = T\tau^0 + \tau$ be the Carlitz module. Then $C$ is a rank 1 Drinfeld module over any $A\dash$field, and if $\gens{f} = \mfp$ is prime in $A$ for $f$ monic the $C$ is height 1 at $\mfp$.
## 4.7 Onward
- There is a theory of morphisms of Drinfeld modules, parallels elliptic curves
- isomorphism iff degree zero,
- if $G/F$ is any algebraically closed extension then $\hom_{\bar F}(\psi,\psi) \injects \Hom_G(\phi, \psi)$,
- $\Endo_F(\psi)$ is commutative and torsionfree projective $A\dash$module of rank $\leq d^2$
- Characterization of when certain group schemes are kernels of isogenies
- Let $E/k \subseteq C_\infty$ be a finite extension, then $E$ is a $\CM_\infty\dash$field iff it contains exactly one prime above $\infty$.
- Can constructed Drinfeld modules with CM by the maximal order $\OO\da A_E$, the $A\dash$integers in $E$ where $\OO\injects \Endo(\psi)$
- For $L\subset C_\infty$ a lattice for $\OO$ of rank $d_1$, it induces an $A\dash$lattice of rank $d\da d_1\cdot [E:k]$ which induces a Drinfeld module $\psi$; say $\psi$ has sufficiently many CMs iff $d_1 = 1$ iff $L\cong I$ where $I$ is an $\OO\dash$ideal.
- Can define conductors: letting $\tilde \OO$ be the maximal order (the $A\dash$integers), the conductor is the largest ideal $\mathfrak c\normal \tilde \OO$ which is also an ideal in $\OO$.
- There is a theory of (potentially) stable reduction and (potentially) good reduction at $v$ a nontrivial discrete valuation on an $A\dash$field $F$.
- Thm 4.10.5: analog of Ogg-Neron-Shaferevich for abelian varieties.
- Analog of the Tate module (a $\mfp\dash$adic Tate module of $\phi$)
- There's an analog of the Tate conjecture for Drinfeld modules (see 10.3)
- There is an analog of Tate-Honda theory for Drinfeld modules
- Analog of supersingular Drinfeld modules
## Ch.5
- Drinfeld modules $\leadsto$ elliptic curves, $T\dash$modules $\leadsto$ abelian varieties.
- For $E$ a $T\dash$module, there is a notion of torsion points for $f\in A = \FF_r[T]$: $E[f] = \ts{e\in E(L) \st f\cdot e = 0}$ where $E(L)$ are the $L\dash$valued points of $E$.
## Ch. 6
- $k$ is a global field over the finite field $\FF_r$
- $A$ is the subring of functions regular away from a fixed place $\infty$
- $T\dash$modules correspond to representing $A$ as a ring of operators on $\GG_a\cartpower{d}$ for some $d$, and a shtuka is a proper module for this action (a locally free sheaf on the complete curve $X$ corresponding to $k$)
- Shtukas connect certain subrings of $L\ts{\tau}$ and certain coherent sheaves on complete curves.
- Can cook up abelian extensions from "division points of sign-normalized rank 1 Drinfeld modules"
## Zeta Functions
- For $X\in \Sch$, define an Euler product $$\zeta_X(s) \da \prod_{x\in \abs X} {1\over 1 - \norm{x}^{-s} },\qquad \norm{x} \da \size \kappa(x) \da \size(\OO_{X, x}/\mfm_x)$$
- Can expand in a sum $$\zeta_X(s) = \sum_{D\in \Div_{>0}(X)} {1\over \norm{D}^{-s} },\qquad \norm{D} \da r^{\deg D}$$ where $\Div_{>0}(X)$ are positive divisors.
- Set$u\da r^{-s}$ to define $Z_X(u) \da \zeta_X(s)$ and $$Z_X(u) = {P_X(u) \over (1-u)(1-ru)}, \qquad \deg P_X(u) = 2g(X)$$ where $P_X(1) = h(k)$ is the class number of the field $k$.
- Satsifies a functional equation $$P_X(u) = r^g u^{2g} P_X\qty{1\over ru}$$, and setting $\xi_X(s) = r^{s(g-1)} Z_X(s)$ one has $$\xi_X(s) = \xi_X(1-s)$$
- $L\dash$series: $X$ a curve over $\FF_r$ where $r= p^{m_0}$, fixed closed point $\infty$, function field $k$ with completion $K$ at $\infty$, $\spec A = X\smts{\infty}$, $C_\infty =$ complete $k$, take algebraic closure, take completion. $G = \Gal(k\sep/k)$, $\ell\neq p\in \ZZ$ a prime, $\rho: G\to \GL(V)$ a continuous finite dimensional $\ell\dash$adic representation.
- E.g. for $Z$ an abelian variety over $k,$ define $\ell\dash$adic Tate module $T_\ell(Z)$ and set $$V \da \Hom_{\ZZladic}(T_\ell(Z), \QQladic) \cong H^1_\et(Z; \QQladic) \in \mods{G}$$
- For $w\in \abs{X}$, define $$f_w(u) \da f_{\bar w}(u) \da \det\qty{1 - \Frob_{\bar w}u \st V^{I_{\bar w}}}$$ where $\bar w$ is an point over $w$ in $\kbar$ and $I_{\bar w}$ is its inertia group and $V^{I_{\bar w}}$ is the fixed subspaces with $\Frob_{\bar w}(u) \in D_{\bar w}/I_{\bar w}$ the geometric Frobenius (inverse of usual).
- Define $$L_V(s) \da \prod_{w\in \abs X} f_w(\norm{w}^{-s})\inv,\qquad \norm{w} \da \size \kappa(w)$$
- Applications: a $p\dash$primary Galois component of the class group is nontrivial $\iff$ a certain classical $L\dash$value is not divisible by $p$ $\iff$ some "finite characteristic" special $\zeta\dash$value is not divisible mod $p$ modulo $\mfp\normal A$. So divisibility of $p$ by $\mfp$ determines nontrivializaty of the $p\dash$primary Galois component.
- Thm 8.18.8 (Yu): let $A = \FF_r[T]$ and $\zeta_A(s)$ be its zeta function, then $\zeta_A(i)$ is transcendental over $k$ for all positive $i$. If $i$ is not divisible by $r-1$, then $\zeta_A(i)/\eps^i$ is transcendental over $k$.
- Here $\zeta_A(s) = \sum_{I\subseteq A} I^{-s}$.
- Recalling classical RH: define $$\xi(s) \da \zeta_\infty(s)\cdot \zeta(s), \qquad \zeta_\infty(s) \da \Gamma\qty{s\over 2} \pi^{-{s\over 2}}$$ and $$\hat\xi(s) \da s(1-s)\xi(s)$$ to get an entire function with $\hat\xi(s) = \hat\xi(1-s)$.
- RH: the zeroes of $\hat\xi$ are of the form ${1\over 2} + i\beta$ for $\beta\in \RR$; can shift and rotate to get a function $\theta(u)$ and equivalently ask that the zeros of $\theta$ are real.
- Relate to a theorem of Wan 8.24.5: Let $y\in \ZZpadic$ where each $r\dash$adic digit of $-y$ is less than $p$ or equal to $r-1$. Then the zeros of $\zeta_A(x, y)$ are in $K$ and simple.
- $\RR$ is a "small" local field and the zeros of $\theta(u)$ are either in $\RR$ or $\CC$, a degree 2 extension. Compare to zeros of $\zeta_A(s)$ which could be in an extension of $K$ of arbitrary (even infinite) degree but are in $K$ itself.