# Monday March 2nd


:::{.theorem title="C., Xarles"}
For $K$ a $p\dash$adic field with residue field $\FF_q$ and $A_{/k}$ a $g\dash$dimensional abelian variety,

a. If $A_{/k}$ has potentially good reduction, then
$$
\# A(k)[\tors]' \leq \floor{ q + 2\sqrt q + 1}^g
$$

b. If $A_{/k}$ has anisotropic reduction, then
$$
\# A(k)[\tors]' \leq 2^{2 \alpha} (q+1)^\mu \floor{q + 2q + 1}^\beta
$$
where for the Neron special fiber satisfies $\alpha + \mu + \beta = g$,

- $\alpha$ is the unipotent rank
- $\mu$ is the toric rank
- $\beta$ is the abelian rank

:::


:::{.example title="?"}
For $E_{/k}$ with additive reduction, so $\# \Phi \leq 4$.
The theorem implies $\# E(k)[\tors]' \leq 2^{2\cdot 1} = 4$, and all possibilities for groups of size at most 4 occur.

> Note that in the paper, there is a bound that doesn't involve the prime, which is much more complicated. 
> The next line is missing a 2 in the $2^{2\alpha}$ bit, which is a typo.

:::


:::{.remark}
Part (b) implies part (a), and when the reduction is potentially good but not good, the bound is actually **better**.
Note for good reduction, $\alpha = \mu = 0$, which recovers the first bound.
For $q=2$, $\floor{2+\sqrt 2 + 1} = 5$, and $2^{2\alpha} = 4^\alpha < 5^\alpha$.
So $(q+1)^\mu \leq \floor{q + 2\sqrt q + 1}^\mu$, and thus
$$
2^{2\alpha}(q+1)^\mu \floor{q + 2q + 1}^\beta 
\leq \floor{q + 2\sqrt q + 1}^{\alpha + \mu + \beta = g}
,$$
thus this bound is sharp and an improvement.

:::

## Base Change and the Neron Model


:::{.remark}
Suppose $k$ is a $p\dash$adic field and $L/K$ a finite extension.
Let $R$ be a valuation ring and let $S$ be the valuation ring of $L$.
There are then Neron models $A_R, A_S$ respectively.
:::


:::{.warnings}
The base change \[
A_R \fiberprod{\spec R} \spec S \to A_S
\]
is not necessarily an isomorphism.
It is an isomorphism if $L/K$ is unramified.
Upon base change, $\alpha$ can only decrease, and $\mu, \beta$ can only increase.

> See Grothendieck's semistable ??: Make a finite base extension with no unipotent reduction?

:::



:::{.definition title="Torii and character groups"}
Let $k$ be a perfect field, then a **torus over $k$** is a linear algebraic group $T_{/k}$ satisfying 
\[
T_{/\bar k} \cong {\GG_m^g}_{/ T_k}
.\]
The **character group** is defined by 
\[
X(T) \definedas \Hom_{\modsleft{g_k}}\qty{ T_{/\bar k}, {\GG_m}_{ / \bar k} } \in \modsleft{g_k}
.\]
For $f$ in this hom, define an action $\sigma \cdot f \definedas (\sigma \circ f \circ \sigma\inv)$.
:::


:::{.remark}
Note that $\sigma \cdot f = f$ iff $\sigma f(x) = f(\sigma x)$, i.e. $f$ is equivariant wrt $\sigma$.

Moreover, $X(T)$ is a $g_k\dash$module with underlying $\ZZ\dash$module $\hom(\GG_m^g, \GG_m) \cong \ZZ^g$.
This uses a fact about algebraic groups.
Thus $T\mapsto X(T)$ gives a contravariant equivalence from the category of (discrete) algebraic torii over $k$ to the category of $g_k\dash$modules with finitely generated free underlying $\ZZ\dash$module.

We are thus reduced to studying $g_k\dash$module structures on $\ZZ_n$, i.e. maps $g_k \to \GL(g, \ZZ)$ with finite image.
We can consider $X^0(T) \definedas X(T) \tensor_{\ZZ} \QQ$, and all goes through as before except now the underlying module is over a field.


:::


:::{.remark}
Isogeny category: categorical localization at isogenies, which can be defined for tori and abelian varieties.
The isogeny category of torii is anti-equivalent to discrete $g_k\dash$modules with underlying $\ZZ\dash$modules a finite-dimensional $\QQ\dash$vector space, and this category is semisimple.

The idea here is that for $0 \to T_1 \to T_2 \to T_3 \to 0$, it isn't true that $T_2 \cong T_1 \oplus T_3$ as an isomorphism, but is an equivalence up to isogeny.

For $T_{/k}$ a torus, we can take the invariants 
\[
X(T)^{g_k} = \{X \in X(T) \mid \sigma \cdot X = X ~\forall \sigma \in g_k\}
\]
the largest submodule on which $g_k$ acts trivially.
Similarly, we can take the coinvariants 
\[
X(T)_{g_k} = X(T) / \theset{\sigma X - \sigma}
,\]
the largest submodule on which $g_k$ acts trivially.

Thus the inclusion $X(T)^{g_k} \injects X(T)$ corresponds to $T \surjects T_S$, the maximal split quotient torus.
Similarly, $X(T) \surjects X(T)_{g_k}$, where we choose a maximal free $\ZZ\dash$submodule, corresponds to a maximal split torus $T_S \injects T$.
Thus $T_S \sim T^S$ are isogenous.

There is thus a maximal split torus, well-defined up to isogeny, and so we can define the **split toric rank of T** as $\dim(T_S) = \dim(T^S)$.
If the split rank is zero, then $T$ is anisotropic (i.e. not even one copy of $\GG_m$ embeds into it).

> Always study algebraic torii by reducing to character groups.
> Define character group for abelian varieties by defining it for its Neron model, which reduces to defining it for the torii.

:::



:::{.theorem title="C., Xarles"}
If $E/\QQ$ is an elliptic curve with everywhere ?? (it does not have split multiplicatively reduction at any $p$.
Then $E(\QQ) [\tors]$ is $\ZZ/n\ZZ$ for $n\in \{1,2,3,4,6\}$, or $(\ZZ/2\ZZ)^2$.
Conversely, all such groups arise already from $E/\QQ$ with $j=0, 1728$.

:::

:::{.remark}
These $j$ invariant pick out curves with certain complex multiplication?
If reduction is multiplicative, it is not potentially good.
Mazur: classified torsion for all elliptic curves over $\QQ$ in the 70s?
\

Next time: $p\dash$adically uniformized something.
:::