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 \(K\),
\[\delta: K \to K, \quad f \mapsto f'\]
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 \(J^nA\), and study algebraic group homomorphisms \(J^nA\to \mathbb{G}_a\). 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
\(\delta(a+b)=\delta(a)+\delta(b) + e_p(a,b)\)
\(\delta(ab) = \delta(a)b^p + a^p\delta(b) + p\delta(a)\delta(b)\)
\(\delta(1)=0, \delta(0)=0\).
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
\[\phi(a) = a^p + p\delta(a)\]
Consider the category of \(\delta\)-rings, we have a forgetful functor
\[\delta\text{-Ring} \to \text{Ring}: \quad (A,\delta) \mapsto A\]
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 \(W_n(B)=\prod_{i=0}^n B\),
\[W_n(B) \xrightarrow{} \prod_{i=0}^n B\] \[(x_0,\dots,x_n)\mapsto \left\langle x_0^{p^n}, x_0^{p^n}+px_1^{p^{n-1}}, \dots, x_0^{p^n}+px_1^{p^{n-1}}+\cdots+p^{n-1}x_{n-1}^p+p^nx_n \right\rangle\]
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
\((x_0,\dots,x_n) + (y_0,\dots,y_n) = (x_0+y_0,x_1+y_1+e_p(x_0,y_0),\dots)\)
\((x_0,\dots,x_n)\cdot (y_0,\dots,y_n) = (x_0y_0, x_0^py_1+x_1y_0^p, \dots)\)
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 \(A=\mathbb{Z}[x]\), \(\phi(x)=x^p+p\cdot(*)\) where \((*)\) 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
\[W_{n+1}\xrightarrow{T,F} W_n(B)\]
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
\[J^nX(B) := X(W_n(B))\]
for any \(R\)-algebra \(B\).
By worker of Borger 2011, this functor \(J^nX\) is representable.
Buium defined arithmetic jet spaces over \(p\)-adic formal schemes. Let \(X=\mathrm{Spec}C\), \(C=R[x\dots]/(f)\), take
\[J^nX = \mathrm{Spec}(J_n C), \quad J_nC = R[x_i^{(j)}]/(f, \delta(f),\dots,\delta^n(f))\]
Diophantine Applications
Explicit Manin-Mumford Bound
Let \(R=\widehat{\mathbb{Z}_p}^{ur}=W(\overline{\mathbb{F}_p})\mathrel{\circlearrowright}_\phi\), \(\delta = \frac{\phi(x) - x^p}{p}\), \(X/R\) a curve with \(g(X)\ge 2\), consider
\[\#\{X(R)\cap \mathrm{Jac}_{tor}(R)\} \le p^{4g}\cdot 3^g \cdot (p(2g-2)+6g)\cdot g!\]
Coleman’s ramified torsion points (\(X\) has good reduction at \(p\) and \(p\ge 2g\)).
\[X(\overline{\mathbb{Q}})\cap \mathrm{Jac}_{tor}(\overline{\mathbb{Q}})\le p^{4g}\cdot 3^g\cdot (p(2g-2)+6g)\cdot g!\]
Part 2
With Netan we have improved this, let \(X/R\) of genus \(g\ge 2\)
\[\Gamma \le J(R)\text{ finite rank.}\]
If \(\mathrm{rank}(\Gamma)<g-1, p\ge 2g\)
\[X(\overline{\mathbb{Q}_p})\cap \Gamma < p^{3g+\mathrm{rank}(\Gamma)}3^g(p(2g-2)+6g)g! + 4g-4\]
Part 3
Let \(f\) be a new form of weight \(2\) curve of level \(\Gamma_0(N)\)
\[\text{CM}\subset X_1(N)\to X_0(N)\to A_f\to A \]
which composes to \(\Phi:X_1(N)\to A\), then \(\Gamma\le A(\overline{\mathbb{Q}})\),
\[\# \Phi(\text{CM})\cap \Gamma <\infty.\]
Application In \(p\)-adic Hodge Theory
Let \(R=W(k)\), \(k\) finite field, \(A/R\) an abelian scheme. For each \(n\) consider
\[0\to N^n \to J^n\xrightarrow{n} A\to 0\]
Apply the functor \(\mathrm{Hom}(,\widehat{\mathbb{G}_a})\) to get
\[0\to \mathrm{Hom}(A,\widehat{\mathbb{G}_a})\to \mathrm{Hom}(J^n,\widehat{\mathbb{G}_a})\to \mathrm{Hom}(N^n,\widehat{\mathbb{G}_a}) \xrightarrow{\delta} \mathrm{Ext}^1(A,\widehat{\mathbb{G}_a})\to \cdots\]
Given any \(\psi\in \mathrm{Hom}(N^n,\mathbb{G}_a)\),
\[0\to \mathbb{G}_a\to \mathrm{Lie}(A_\psi^*)\to \mathrm{Lie}(A)\to 0\]