Author: Eiko

Time: 2025-02-25 11:04:03 - 2025-02-25 11:04:03 (UTC)

Arithmetic Jet Spaces

• Motivation

• Arithmetic Jet Spaces

• Some Diophantine Applications

• Application In $p$-adic Hodge Theory

Motivation

Around the 1960s, let $K$ be a function field of one variable, for example $K=\mathbb{C}(t)$, we can take a derivation on $K$,

$$\delta :K\to K,f\mapsto f^{\prime }$$

that gives you a differential algebra. Around the 1960s, Manin studied this function; he considered abelian $A/K$, differential characters $A(K)\to (K,+)$, giving him a proof of Mordell-Lang over $K$.

Later, Buium proved an explicit bound by 1993 on bounds on function fields of characteristic $0$.

Buium-Voloch (1996) proved Mordell-Lang over function fields of positive characteristic.

Around 1995, he introduced $p$-derivations and proved Mordell-Lang over number fields.

Construct algebraic groups $J^{n}A$, and study algebraic group homomorphisms $J^{n}A\to {\mathbb{G}}_a$. We will discuss these in arithmetic settings.

$p$-adic Derivations

Let $p$ be a fixed prime, define $p$ derivation

Let $A$ be a ring, a $p$-derivation on $A$ is a set theoretic map $\delta :A\to A$ such that

1. $\delta (a+b)=\delta (a)+\delta (b)+e_p(a,b)$

2. $\delta (ab)=\delta (a)b^p+a^p\delta (b)+p\delta (a)\delta (b)$

3. $\delta (1)=0,\delta (0)=0$.

where $e_p(x,y)=\frac{x^p+y^p-(x+y{)}^p}{p}$ is a polynomial in $\mathbb{Z}[x,y]$.

We say $(A,\delta )$ is a $\delta$-ring.

• $\delta$-ring $\Rightarrow$ lift of Frobenius

  $$\varphi (a)=a^p+p\delta (a)$$

• Consider the category of $\delta$-rings, we have a forgetful functor

  $$\delta \text{-Ring}\to \text{Ring}:(A,\delta )\mapsto A$$

  that forgets the $\delta$-structure. It has an adjoint functor $p$-typical Witt vectors

  $$A\to W(A)$$

Witt Vectors

Let $B$ be any ring and $n\in \mathbb{N}$. Construct $W_n(B)=\prod _{i=0}^{n}B$,

$$W_n(B)\stackrel{}{\to }\prod _{i=0}^{n}B$$ $$(x_0,\dots ,x_n)\mapsto ⟨x_0^{p^n},x_0^{p^n}+p{x}_1^{p^{n-1}},\dots ,x_0^{p^n}+p{x}_1^{p^{n-1}}+\cdots +p^{n-1}{x}_{n-1}^p+p^n{x}_n⟩$$

There exists unique ring structure such that $W_n(B)$ is a becomes a natural transformation of functors $W_n\Rightarrow \prod _{0\le i\le n}$ on $\mathbf{Ring}\to \mathbf{Ring}$.

This ring structure looks like

• $(x_0,\dots ,x_n)+(y_0,\dots ,y_n)=(x_0+y_0,x_1+y_1+e_p(x_0,y_0),\dots )$

• $(x_0,\dots ,x_n)\cdot (y_0,\dots ,y_n)=(x_0{y}_0,x_0^p{y}_1+x_1{y}_0^p,\dots )$

• Having a $\delta$-structure on $A$ is equivalent to having a ring homomorphism $A\to W_1(A)$

Example

• Let $A=\mathbb{Z}$, by Fermat’s Little Theorem, we have

  $$\delta (a)=\frac{a-a^p}{p}\in \mathbb{Z}$$

  where we chose a lifted Frobenius to be $\mathrm{id}$.

• Let $A=\mathbb{Z}[x]$, $\varphi (x)=x^p+p\cdot (\ast )$ where $(\ast )$ can be any polynomial and it will give you a lifted Frobenius.

Arithmetic Jet Spaces

$B={\mathbb{F}}_p$, $W_n(B)\cong \mathbb{Z}/p^{n+1}\mathbb{Z}$

• In general this Witt vectors have a function

  $$W_{n+1}\stackrel{T,F}{\to }{W}_n(B)$$

  called truncation $T$ and a Frobenius function $F$.

• Verschibung : $V:W_n(B)\to W_{n+1}(B)$

• Teichmuller lift : $[\cdot ]:B\to W_n(B)$ map of multiplicative monoid.

Let $(R,\delta )$ be a $\delta$-ring, If $B$ is an $R$-algebra, then $W_n(B)$ is an $R$-algebra.

• Definition. Fix a $\delta$-ring $R$. Let $X/R$ be a scheme, for each $n\ge 0$, you can define a functor

  $$J^{n}X(B):=X(W_n(B))$$

  for any $R$-algebra $B$.

By worker of Borger 2011, this functor $J^{n}X$ is representable.

Buium defined arithmetic jet spaces over $p$-adic formal schemes. Let $X=\mathrm{Spec}C$, $C=R[x\dots ]/(f)$, take

$$J^{n}X=\mathrm{Spec}(J_{n}C),J_{n}C=R[x_i^{(j)}]/(f,\delta (f),\dots ,{\delta }^n(f))$$

Diophantine Applications

Explicit Manin-Mumford Bound

Let $R={\widehat{{\mathbb{Z}}_p}}^{ur}=W(\overline{{\mathbb{F}}_p}){\circlearrowright }_{\varphi }$, $\delta =\frac{\varphi (x)-x^p}{p}$, $X/R$ a curve with $g(X)\ge 2$, consider

$$\mathrm{\#}\{X(R)\cap {\mathrm{Jac}}_{tor}(R)\}\le p^{4g}\cdot 3^g\cdot (p(2g-2)+6g)\cdot g!$$

Coleman’s ramified torsion points ($X$ has good reduction at $p$ and $p\ge 2g$).

$$X(\bar{\mathbb{Q}})\cap {\mathrm{Jac}}_{tor}(\bar{\mathbb{Q}})\le p^{4g}\cdot 3^g\cdot (p(2g-2)+6g)\cdot g!$$

Part 2

With Netan we have improved this, let $X/R$ of genus $g\ge 2$

$$\mathrm{\Gamma }\le J(R)\text{ finite rank.}$$

If $\mathrm{rank}(\mathrm{\Gamma })<g-1,p\ge 2g$

$$X(\overline{{\mathbb{Q}}_p})\cap \mathrm{\Gamma }<p^{3g+\mathrm{rank}(\mathrm{\Gamma })}{3}^g(p(2g-2)+6g)g!+4g-4$$

Part 3

Let $f$ be a new form of weight $2$ curve of level ${\mathrm{\Gamma }}_0(N)$

$$\text{CM}\subset X_1(N)\to X_0(N)\to A_f\to A$$

which composes to $\mathrm{\Phi }:X_1(N)\to A$, then $\mathrm{\Gamma }\le A(\bar{\mathbb{Q}})$,

$$\mathrm{\#}\mathrm{\Phi }(\text{CM})\cap \mathrm{\Gamma }<\mathrm{\infty }.$$

Application In $p$-adic Hodge Theory

Let $R=W(k)$, $k$ finite field, $A/R$ an abelian scheme. For each $n$ consider

$$0\to N^n\to J^n\stackrel{n}{\to }A\to 0$$

Apply the functor $\mathrm{Hom}(,\widehat{{\mathbb{G}}_a})$ to get

$$0\to \mathrm{Hom}(A,\widehat{{\mathbb{G}}_a})\to \mathrm{Hom}(J^n,\widehat{{\mathbb{G}}_a})\to \mathrm{Hom}(N^n,\widehat{{\mathbb{G}}_a})\stackrel{\delta }{\to }{\mathrm{Ext}}^1(A,\widehat{{\mathbb{G}}_a})\to \cdots$$

Given any $\psi \in \mathrm{Hom}(N^n,{\mathbb{G}}_a)$,

$$0\to {\mathbb{G}}_a\to \mathrm{Lie}(A_{\psi }^{\ast })\to \mathrm{Lie}(A)\to 0$$