Problem # 1:
Let \(R\) be a ring, let \(\gp\) be a prime ideal of \(R\),
and let \(R_\gp\) denote the completion of \(R\) at \(\gp\), i.e.,
\[
R_\gp = \lim_{\leftarrow} R/\gp^{n+1}.
\]
Problem # 2:
For \(x\in R_\gp\), define
\[
\ord_\gp(x) = \min \{ n \ge 0 : x \not\equiv 0\pmod{\gp^{n+1}} \}.
\]
(And set \(\ord_\gp(0)=\infty\).)
Prove that:
Problem # 3:
(a)
Prove that \(R_\gp\) is complete, i.e., prove that every Cauchy
sequence in \(R_\gp\) converges to an element of \(R_\gp\).
Problem # 4:
This exercise gives two generalizations of Hensel's lemma.
The first relaxes the condition that \(f'(a_0)\not\equiv0\pmod{\gp}\),
and the second is an algebraic geometry version that is very useful.
(If you haven't taken algebraic geometry, don't worry about the second
version.)
Problem # 5:
We defined \(|\cdot|\) to be non-archimedean
if
\[
|x+y| \le \max \{ |x|, |y| \}
\quad\hbox{for all \(x\) and \(y\)}.
\]
And we defined \(|\cdot|\) to be archimedean
if for all \(x\ne0\) and all \(C>0\), there is an integer \(n\ge1\)
such that \(|nx|>C\). Prove that these definitions are consistant
in the sense that \(|\cdot|\) is non-archimedean if and only if
it is not archimedean.
Problem # 6:
Let \(R\) be the ring of integers in a number field \(K\), let \(\gp\)
be a prime ideal, let \(q=N_{K/\mathbb{Q}}\gp\),
and for \(x\in K^*\) define
\[
|x|_\gp = q^{-\ord_\gp(x)}.
\]
(Here \(\ord_\gp(x)\) is defined by writing \(x=a/b\) with \(a,b\in R\)
and setting \(\ord_\gp(x)=\ord_\gp(a)-\ord_\gp(b)\).) We also set
\(|0|_\gp=0\).
Problem # 7:
We proved in class that every non-archimedean absolute value on
\(\mathbb{Q}\) is equivalent to some \(p\)-adic absolute value
\(|\cdot|_p\). Let \(|\cdot|\) be an archimedean absolute value on
\(\mathbb{Q}\). Prove that \(|\cdot|\) is equivalent to the usual
absolute value \(|x|_\infty=\max\{x,-x\}\) on \(\mathbb{Q}\)
associated to the embedding \(\mathbb{Q}\to\mathbb{R}\). (Hint:
Fix an integer \(b\gt 1\) and use the \(b\)-expansion of
\(a\in\mathbb{Z}\), i.e., write \(a=c_0+c_1b+\cdots+c_rb^r\) with
\(0\le c_i\lt b\).)
Problem # 8:
If you've never done this type of construction before, it's worth
doing once.
Problem # 9:
Let \((K,|\cdot|)\) be a valued field and let
\[
f(X) = X^d+a_1X^{d-1}+\cdots+a_d
= (X-b_1)(X-b_2)\cdots(X-b_d)
\]
be a polynomial, where \(a_1,\ldots,a_d,b_1,\ldots,b_d\in K\).
Let \(C=2\) if the absolute value is archimedean and let \(C=1\)
if the absolute value is non-archimedean. Prove that
\[
C^{-d} |f|
\le \prod_{i=1}^d \max\{ |b_i|,1 \}
\le C^d |f|.
\]
This gives a very useful relation between the size of the coefficients
of a polynomial and the size of its roots.
Problem # 10:
(a)
Prove that the natural inclusion
\[
R_\gp := \lim_{\leftarrow} R/\gp^{n+1}
\subset
\prod_{n\ge 0} R/\gp^{n+1}
\]
makes the inverse limit
into a closed subset of the product, where each \(R/\gp^{n+1}\)
is given the discrete topology, and where the product is given the product
topology. Deduce that \(R_\gp\) is compact.
(We are assuming that the \(R/\gp^{n+1}\) are finite).
Problem # 11:
(a) Find all extensions \(K/\mathbb{Q}_p\) satisfying
\([K:\mathbb{Q}_p]=2\). (The answer is more complicated
for \(p=2\) than it is for odd primes.)
Important Note:
The rest of the course will cover material in
The Arithmetic of Elliptic Curves, 2nd edition.
Exercises in the book will be denoted [AEC C.N], where C is the
chapter number and N is the problem number.
Problems # 12–15:
Problems on Formal Groups (AEC page 135)
Problem # XXX16:
Let \(\mathcal{F}\) and \(\mathcal{G}\) be formal groups over a ring \(R\),
and let \(\text{Hom}_R(\mathcal{F},\mathcal{G})\) denote the set of
formal-group homomorphisms from \(\mathcal{F}\) to \(\mathcal{G}\).
Problem # XXX17:
Prove the following generalization
of Lemma IV.2.4. Let \(R\) be a ring,
let \(x\) be an indeterminate, and let \(f(T)\in R[x][[T]]\) be a power
series with coefficients in the polynomial ring \(R[x]\). Suppose
further that \(f\) has the form
\[
f(T) = xT + (\text{higher order terms}).
\]
Prove that there is a unique power series \(g(T)\in R[x,x^{-1}][[T]]\)
such that \(f(g(T))=T\). More precisely, prove that \(g(T)\) has the form
\[
g(T) = \sum_{n=1}^\infty \frac{b_n}{x^{2n-1}}T^n
\quad\text{with \(b_n\in R[x]\).}
\]
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(a)
Prove that \(R_\gp\) is an integral domain.
(b)
There are natural maps \(\alpha_n : R_\gp \to R/\gp^{n+1}\). Prove that
\(\operatorname{ker}(\alpha_0)\) is the unique maximal ideal of \(R_\gp\).
In particular, \(R_\gp\) is a local ring.
By abuse of notation, we will often write
\(\gp\) for \(\operatorname{ker}(\alpha_0)\), although more accurate
notations would be \(\gp R_\gp\) or \(\gp_\gp\).
(a)
\(\ord_\gp(xy) = \ord_\gp(x)+\ord_\gp(y)\).
(b)
\(\ord_\gp(x+y) \ge \min\{ \ord_\gp(x),\ord_\gp(y) \}\).
(c)
If \(\ord_\gp(x)\ne\ord_\gp(y)\), then
\(\ord_\gp(x+y) = \min\{ \ord_\gp(x),\ord_\gp(y) \}\).
(b)
Prove that \(R_\gp\) is Hausdorff.
(c)
If \(R/\gp\) is finite, prove that \(R_\gp\) is compact, and deduce
that \(K_\gp\) is locally compact.
(a)
Suppose that \(f(X)\in R_\gp[X]\) is a non-zero polynomial
and that \(a_0\in R\) satisfies
\[
f(a_0)\equiv 0\pmod{\gp^{2k+1}}
\quad\hbox{and}\quad
f'(a_0)\not\equiv 0\pmod{\gp^{k+1}}
\]
for some \(k\gt 0\).
Prove that there is a unique \(a\in R_\gp\) satisfying
\[
a\equiv a_0\pmod{\gp^{2k+1}}
\quad\hbox{and}\quad
f(a)=0.
\]
(b)
Let \(F(X_1,\ldots,X_N)\in R_\gp[X_1,\ldots,X_N]\)
be a non-zero polynomial and let \(V\) be the affine variety defined
by \(F=0\). Also let \(\tilde V\) denote the variety \(F=0\) over the
residue field \(R_\gp/\gp\). Let \(Q_0\in \tilde V(R_\gp/\gp)\) be
a non-singular point of \(\tilde V\). Prove that there is a
point \(Q\in V(R_\gp)\) satisfying \(Q\equiv Q_0\pmod{\gp}\). Is \(Q\)
necessarily unique?
(b')
Prove a more general result that holds
for a mod \(\gp\) nonsingular point on any affine variety defined
by equations with coefficients in \(R_\gp\).
(b'')
Prove a simlar result that holds
for a mod \(\gp\) nonsingular point on any projective variety defined
by homogeneous equations with coefficients in \(K_\gp\).
(a)
Prove that \(|\cdot|_\gp\) is a non-archimedean absolute value on \(R\).
(b)
Prove that \(|\cdot|_\gp\) extends uniquely to a non-archimedean
absolute value on \(R_\gp\).
(c)
Prove that \(R_\gp\) is complete for the topology induced
by the absolute value \(|\cdot|_\gp\).
Let \((K,|\cdot|)\) be a valued field, i.e., a field with an absolute
value on it. Prove that there exists a unique (up to unique isomorphism)
valued field \((K',|\cdot|')\) with the following properties:
(i)
\((K',|\cdot|')\) is complete, i.e., every Cauchy sequence converges.
(ii)
There is an inclusion \((K,|\cdot|)\subset(K',|\cdot|')\) of valued
fields.
(iii)
If \((K,|\cdot|)\subset(K'',|\cdot|'')\) is any other inclusion of
valued fields with \(K''\) complete, then there is a unique
inclusion \((K',|\cdot|')\subset(K'',|\cdot|'')\) that is compatible
with the inclusions of \((K,|\cdot|)\).
The field \(K'\) is called the completion of \(K\). Prove further
that \(K\) is dense in \(K'\) with respect topology induced by the
absolute value.
Hint: To construct \(K'\), consider the collection
\(\mathcal{C}\) consisting of all Cauchy sequences in \(K\), and mod
out \(\mathcal{C}\) by the relation that \((a_i)\sim(b_i)\) if \(\lim
|a_i-b_i|=0\).
u8
(b)
The unit group \(R_\gp^*\) is given a topolgy via the embedding
\[
R_\gp^* \longrightarrow R_\gp\times R_\gp,\qquad
x \longmapsto (x,x^{-1}).
\]
Prove that \(R_\gp^*\) is compact.
(b)
Same question for \([K:\mathbb{Q}_p]=3\), at least
for \(p\ne 3\). Can you also do \(p=3\)? Try doing the
cubic Galois extensions first.
# 12: [AEC 4.1]
# 13: [AEC 4.2(a)]
# 14: [AEC 4.3]
# 15: [AEC 4.6]
(a)
Define a binary operation \(\star\) on
\(\text{Hom}_R(\mathcal{F},\mathcal{G})\) as follows: for
\(f_1,f_2,\in \text{Hom}_R(\mathcal{F},\mathcal{G})\), set
\[
(f_1\star f_2)(T) = G(f_1(T),f_2(T)).
\]
Prove that \(f_1\star f_2\) is in \(\text{Hom}_R(\mathcal{F},\mathcal{G})\),
and that \(\star\) makes \(\text{Hom}_R(\mathcal{F},\mathcal{G})\) into a
group.
(b)
Let \(\text{End}_R(\mathcal{F})=\text{Hom}_R(\mathcal{F},\mathcal{F})\).
Prove that \(\text{End}_R(\mathcal{F})\) is a ring, where the \(\star\)
operation from (a) is "addition" and composition of power series
is "multiplication".