Exercises

Chapter 19 Exercises

Section 19.1 Set 1

Exercises Exercises

1.

Let $k$ be a finite field of characteristic $p\text{.}$ Define the map

\begin{equation*} \psi_0: \FF_p \to \CC^\times, \qquad x \mapsto e^{2 \pi i x/p}. \end{equation*}

Prove that every homomorphism from (the additive group of) $k$ to $\CC^\times$ has the form $x \mapsto \psi_0(\Trace_{k/\FF_p}(xz))$ for some unique $z \in k\text{.}$

Hint.

$x,y \mapsto \Trace_{k/\FF_p}(x,y)$$\FF_p$$k\text{.}$$k$$x \mapsto (y \mapsto \Trace_{k/\FF_p}(xy))$$k \to \Hom_{\FF_p}(k, \FF_p)$$\Trace_{k/\FF_p}: k \to \FF_p$$q$$k\text{,}$$x \mapsto x + x^p + x^{p^2} + \cdots + x^{q/p}\text{,}$$q$$k\text{.}$

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2.

Let $k$ be a finite field of order $q$ and fix a nontrivial additive character (homomorphism) $\psi: k \to \CC^\times\text{.}$ For $\chi: k^\times \to \CC^\times$ a nontrivial multiplicative character, define the Gauss sum

\begin{equation*} G_\psi(\chi) = \sum_{x \in k^\times} \chi(x) \psi(x). \end{equation*}

Prove that $G_\psi(\chi) G_{\psi^{-1}}(\overline{\chi}) = q\text{,}$ where $\overline{\chi}$ is the character for which $\overline{\chi}(x)$ is the complex conjugate of $\chi(x)\text{.}$

Hint.

Exercise 19.1.1$x,y \in k^\times\text{,}$$x/y\text{.}$

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3.

With notation as in Exercise 19.1.2, verify that $G_{\psi^{-1}}(\chi) = \chi(-1) G_\psi(\chi)\text{.}$

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4.

Fix a choice of $\chi$ as in Exercise 19.1.2. For $P(T) = T^n + P_{n-1} T^{n-1} + \cdots + P_0 \in k[T]$ a monic polynomial, define

\begin{equation*} \lambda(P) = \chi(P_0) \psi(P_{n-1}). \end{equation*}

(In particular, $\lambda(1) = 1\text{.}$) Show that

\begin{equation*} \lambda(P_1 P_2) = \lambda(P_1) \lambda(P_2) \qquad (P_1, P_2 \in k[T]) \end{equation*}

and deduce that for each positive integer $n\text{,}$ in $\CC\llbracket U \rrbracket$ we have

\begin{equation*} \sum_{P \in k[T] \mbox{\scriptsize\, monic}} \lambda(P) U^{\deg(P)} = \prod_{Q \in k[T] \mbox{\scriptsize\, monic irreducible}} (1 - \lambda(Q) U^{\deg{Q}})^{-1}. \end{equation*}

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5.

Show that for $n$ a nonnegative integer,

\begin{equation*} \sum_{P \in k[T] \mbox{\scriptsize\, monic}, \deg(P) = n} \lambda(P) U^{\deg(P)} = \begin{cases} 1 & n=0 \\ G_\psi(\chi) U & n=1 \\ 0 & n > 1. \end{cases} \end{equation*}

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6.

With notation as in Exercise 19.1.5, let $k'$ be an extension of $k$ of degree $v\text{.}$ Let $\psi': k' \to \CC^\times$ be the additive character given by $\psi \circ \Trace_{k'/k}\text{.}$ Given $\chi\text{,}$ let $\chi'$ be the multiplicative character given by $\chi \circ \Norm_{k'/k}\text{.}$ For $P' \in k'[T]$ monic, define $\lambda'$ by analogy with $\lambda\text{.}$ For $P \in k[T]$ monic irreducible, let $P'$ run over the irreducible factors of $P$ in $k'[T]\text{.}$ Prove that

\begin{equation*} \prod_{P'} (1 - \lambda'(P') U^{v \deg(P')}) = \prod_{\rho=0}^{v-1} (1 - \lambda(P) (e^{2\pi i \rho/v} U)^{\deg(P)}). \end{equation*}

Hint.

$-\xi$$P'\text{,}$$k(\xi)/k$$k'(\xi)/k'\text{.}$

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7.

Using all of the above, deduce the Davenport–Hasse relation

\begin{equation*} - G_{\psi'}(\chi') = (- G_\psi(\chi))^v. \end{equation*}

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Section 19.2 Set 2

Throughout, let $\FF_q$ denote a finite field of characteristic $p\text{.}$

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Exercises Exercises

1.

For $X$ an algebraic variety over $\FF_q\text{,}$ we write the zeta function of $X$ as $Z(X, q^{-s})$ for

\begin{equation*} Z(X, T) = \prod_{x \in X^\circ} (1 - T^{\deg(x)})^{-1}, \end{equation*}

where $X^\circ$ denotes the set of Galois orbits of $\overline{\FF_q}$-points and $\deg(x)$ is the cardinality of such an orbit. Prove that in $\QQ \llbracket T \rrbracket\text{,}$ we have the equality

\begin{equation*} Z(X,T) = \exp \left( \sum_{n=1}^\infty \frac{T^n}{n} \#X(\FF_{q^n}) \right). \end{equation*}

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2.

For $X$ equal to the $n$-dimensional projective space over $\FF_q\text{,}$ compute that

\begin{equation*} Z(X,T) = \frac{1}{(1-T)(1-qT)\cdots(1-q^n T)}. \end{equation*}

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3.

Prove that the following statements are equivalent.

The power series $Z(X,T)$ represents a rational function in $T\text{.}$

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There exist $\alpha_1,\dots,\alpha_r, \beta_1,\dots,\beta_s \in \CC$ such that
\begin{equation*} \#X(\FF_{q^n}) = \alpha_1^n + \cdots + \alpha_r^n - \beta_1^n - \cdots - \beta_s^n \qquad (n=1,2,\dots). \end{equation*}

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4.

Let $X$ be the Grassmannian of $k$-dimensional subspaces of $m$-space over $\FF_q\text{.}$

Compute $\#X(\FF_{q^n})\text{;}$ your answer should be a polynomial in $q^n$ depending on $k$ and $m\text{.}$

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Compute $Z(X,T)\text{.}$

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Hint.

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5.

Choose $a_0,\dots,a_r \in \FF_q^\times\text{.}$ For $d$ a positive integer dividing $q-1\text{,}$ let $X_d$ be the projective hypersurface $a_0 x_0^d + \cdots + a_r x_r^d = 0\text{.}$

Let $G_d$ be the group of homomorphisms $\chi: \FF_q^\times \to \CC^\times$ of order $d\text{.}$ For $\chi \in G_d\text{,}$ extend the definition of $\chi$ to $\FF_q$ by setting $\chi(0) = 1$ if $\chi=1$ and $\chi(0) = 0$ otherwise. Show that
\begin{equation*} 1 + (q-1)\#X_d(\FF_q) = \sum_{(u_0,\dots,u_r) \in X_1} \sum_{\chi_0,\dots,\chi_r \in G_d} \prod_{i=0}^r \chi_i(u_i). \end{equation*}

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Show that if $\chi_0,\dots,\chi_r \in G_d$ are neither all equal to 1 nor all distinct from 1, then
\begin{equation*} \sum_{(u_0,\dots,u_d) \in X_1} \prod_{i=0}^r \chi_i(u_i) = 0. \end{equation*}

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Let $T$ be the set of tuples $(\chi_0,\dots,\chi_r) \in G_d \setminus \{1\}$ with $\chi_0 \cdots \chi_r = 1\text{.}$ For $(\chi_0,\dots,\chi_r) \in T\text{,}$ define the Jacobi sum
\begin{equation*} j(\chi_0,\dots\chi_r) = \frac{1}{q-1} \sum_{u_0,\dots,u_r \in \FF_q: u_0 + \cdots + u_r = 0} \chi_0(u_0) \cdots \chi_r(u_r). \end{equation*}
Show that
\begin{equation*} \#X_d(\FF_q) = 1 + q + \cdots + q^{r-1} + \sum_{(\chi_0,\cdots,\chi_r) \in T} \chi_0(a_0^{-1}) \cdots \chi_r(a_r^{-1}) j(\chi_0,\dots,\chi_r). \end{equation*}

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Fix an additive character $\psi: \FF_q \to \CC^\times\text{.}$ Show that
\begin{equation*} j(\chi_0,\dots,\chi_r) = \frac{1}{q} G(\chi_0,\psi) \cdots G(\chi_r, \psi) \end{equation*}
where $G(\chi, \psi)$ denotes the Gauss sum Exercise 19.1.2.

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Keep notation as in Exercise 19.2.5, but assume only that $d$ is not divisible by $p$ (not that it divides $q-1$).

Show that $\#X_d(\FF_q) = \#X_e(\FF_q)$ for $e = \gcd(d, q-1)\text{.}$

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Using the Davenport–Hasse relation, show that the rationality, functional equation, and Riemann hypothesis hold for $Z(X_d, T)\text{.}$

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Section 19.3 Set 3

Throughout, let $\FF_q$ denote a finite field of characteristic $p\text{.}$ Assume the Weil conjectures for curves and abelian varieties unless otherwise specified.

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Exercises Exercises

1.

Let $X$ be a nonzero abelian variety over $\FF_q\text{.}$ Prove that if $q \geq 5\text{,}$ then the group $X(\FF_q)$ is nontrivial.

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2.

Let $X$ be a curve over $\FF_q$ such that $\#X(\FF_q) = 1\text{.}$

If $q=3$ or $q=4\text{,}$ prove that
\begin{equation*} Z(X,T) = \frac{1 - qT + qT^2}{(1-T)(1-qT)}. \end{equation*}

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If $=2\text{,}$ prove that the genus of $X$ is at most 4, and that there are at most 6 possibilities for $Z(X,T)\text{.}$

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Show that each of the 8 possibilities occurs for a unique $X$ up to isomorphism.

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3.

Let $X$ be an abelian variety of dimension $g$ over $\FF_q\text{.}$ Assuming only the existence of complex numbers $\alpha_1,\dots,\alpha_{2g}$ such that

\begin{equation*} X(\FF_{q^n}) = (1-\alpha_1^n)\cdots(1-\alpha_{2g}^n) \qquad (n=1,2,\dots), \end{equation*}

compute $Z(X,T)\text{.}$

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4.

Using the Honda–Tate theorem, prove that if $A_1, A_2$ are abelian varieties over $\FF_q$ and $P_1(A_1, T)$ divides $P_1(A_2, T)\text{,}$ then $A_1$ is isogenous to the product of $A_2$ with some other abelian variety.

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5.

Let $X$ be a curve of genus $g$ over $\FF_q\text{.}$ Prove the following refinement of the Weil bound due to Serre:

\begin{equation*} \left| \#X(\FF_q) - q - 1 \right| \leq g \lfloor 2 \sqrt{q} \rfloor. \end{equation*}

Hint.

$\lfloor 2 \sqrt{q} \rfloor + 1 + \alpha + \overline{\alpha}$$\alpha$

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6.

Let $P(T) = \sum_{i=0}^{2g} a_i T^i$ be a polynomial over $\ZZ$ such that $a_0 = 1\text{,}$ $a_{g+i} = q^i a_{g-i}$ for all $i\text{,}$ all roots of $P(T)$ in $\CC$ lie on the circle $|T| = q^{-1/2}\text{,}$ and in addition $a_g$ is not divisible by $p$ (that is, $P$ is an ordinary Weil polynomial). Use the Honda–Tate theorem to show that $P(T)$ occurs as $P_1(A,T)$ for some abelian variety $A$ over $\FF_q$ (without raising $P$ to a power).

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Section 19.4 Set 4

Exercises Exercises

1.

Let $K$ be a number field. Using the Chebotarëv density theorem, prove that the Frobenius elements corresponding to maximal ideals of $\mathfrak{o}_K$ are dense in the absolute Galois group $G_K\text{.}$

Hint.

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2.

In this exercise, we prove Theorem 11.1.4.

Let $f(T) = \sum_{n=0}^\infty a_n T^n$ be a power series over an arbitrary field $K\text{.}$ Prove that $f(T)$ represents a rational function over $K$ if and only if for some positive integer $m\text{,}$ the determinants of the $(m+1) \times (m+1)$ matrices $A_{n,m} = (a_{n+i+j})_{i,j=0}^m$ vanish for all sufficiently large $n\text{.}$

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Let $f(T) = \sum_{n=0}^\infty a_n T^n$ be a power series over $\ZZ\text{.}$ Let $r>0$ be a real number such that over $\QQ_p\text{,}$ there exists a polynomial $P(T)$ of degree $d\lt m$ such that $P(T)f(T)$ converges for $|T| \lt r+\epsilon$ for some $\epsilon > 0\text{.}$ (We do not assume that $P$ has coefficients in $\ZZ\text{.}$) Prove that for some $C > 0\text{,}$ $\left| \det(A_{n,m}) \right|_p \leq C r^{-n(m-d)}$ for all $n\text{.}$

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Let $f(T) = \sum_{n=0}^\infty a_n T^n$ be a power series over $\ZZ\text{.}$ Let $R$ and $r$ be real numbers with $Rr > 1$ such that over $\CC\text{,}$ $f(T)$ converges for $|T| \lt R\text{;}$ and over $\QQ_p\text{,}$ $f(T)$ is the ratio of two series that converge for $|T| \lt r\text{.}$ Prove that $f$ represents a rational function.

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Hint.

$r$$r-\epsilon$$(R-\epsilon)(r-\epsilon)>1\text{,}$$\left| \det(A_{n,m}) \right|_\infty\text{.}$

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3.

Let $\pi$ be an element of an algebraic closure of $\QQ_p$ satisfying $\pi^{p-1} = -p\text{.}$ (You may use without proof the fact that $\ZZ_p[\pi]$ is a discrete valuation ring with maximal ideal $(\pi)\text{.}$) Define the power series

\begin{equation*} E_\pi(T) = \exp(\pi(T-T^p)) \in \QQ_p(\pi) \llbracket T \rrbracket. \end{equation*}

Prove that $E_\pi(T) \in 1 + \pi \ZZ_p[\pi] \llbracket T \rrbracket\text{.}$

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Prove that $E_\pi(T)$ has radius of convergence strictly greater than 1. In particular, it makes sense to evaluate it at any element of $\ZZ_p[\pi]\text{.}$

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Prove that if $t \in \ZZ_p$ satisfies $t^p = t\text{,}$ then $E_\pi(t)^p = 1\text{.}$

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Hint.

\begin{equation*} E_\pi(T)^p = \exp(\pi p T) \exp(-\pi p T^p) \end{equation*}

$t$separately

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4.

With notation as in Exercise 19.4.3, let $n$ be a positive integer and define

\begin{equation*} E_n(T) \colonequals \exp(\pi(T-T^{p^n})) = E_\pi(T) E_\pi(T^p) \cdots E_\pi(T^{p^{n-1}}) \in \QQ_p(\pi) \llbracket T \rrbracket. \end{equation*}

Show that the formula $t \mapsto E_n([t])$ defines a nontrivial additive character on $\FF_{p^n}\text{,}$ where $[t]$ denotes the unique element of $\ZZ_{p^n}$ (the finite étale extension of $\ZZ_p$ with residue field $\FF_{p^n}$) lifting $t$ and satisfying $t^{p^n} = t\text{.}$

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5.

Set $q = p^n$ and let

\begin{equation*} f = \sum_{I = (i_1,\dots,i_d)} a_I x_1^{i_1} \cdots x_d^{i_d} \in \FF_{q}[x_1,\dots,x_d] \end{equation*}

be a polynomial. Prove that for any positive integer $m\text{,}$ the number of points $(x_1,\dots,x_d) \in (\FF_{q^{m}}^\times)^d$ for which $f(x_1,\dots,x_d) = 0$ equals

\begin{equation*} \frac{(q^{m}-1)^d}{q^{m}} \left(1 + (q^{m}-1) \sum_{x_0,\dots,x_d \in \FF_{q^m}^\times} \prod_{I: a_I \neq 0} \prod_{j=0}^{m-1} E_\pi(a_I ([x_0] [x_1]^{i_1} \cdots [x_d]^{i_d})^{q^{j}}) \right). \end{equation*}

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Section 19.5 Set 5

Exercises Exercises

1.

Define the rings

\begin{equation*} R \colonequals \ZZ[x_1,y_1,x_2,y_2,\dots], \quad R' \colonequals \QQ[x_1,y_1,x_2,y_2,\dots], \quad F \colonequals \Frac(R) = \Frac(R'). \end{equation*}

Define the power series $x = 1 + x_1 T + x_2 T^2 + \cdots, y = 1 + y_1 T + y_2 T^2 + \cdots\text{,}$ and

\begin{equation*} f = 1/\exp(\log(1/x) \star \log(1/y)) \in R' \llbracket T \rrbracket \end{equation*}

where $\star$ denotes the Hadamard product:

\begin{equation*} (a_1 T + a_2 T^2 + \cdots) \star (b_1 T + b_2 T^2 + \cdots) = a_1b_1T + a_2b_2 T^2 + \cdots \end{equation*}

Let $V_1, V_2$ be two finite-dimensional vector spaces over $F$ equipped with endomorphisms $\varphi_1,\varphi_2$ satisfying, for some positive integer $n\text{,}$
\begin{align*} \det(1-\varphi_1 T, V_1)^{-1} &\equiv 1 + x_1 T + \cdots + x_n T^n \pmod{T^{n+1} F \llbracket T \rrbracket}, \\ \det(1-\varphi_2 T, V_2)^{-1} &\equiv 1 + y_1 T + \cdots + y_n T^n \pmod{T^{n+1} F \llbracket T \rrbracket}. \end{align*}
Prove that
\begin{equation*} \det(1 - (\varphi_1 \otimes \varphi_2) T, V_1 \otimes_F V_2)^{-1} \equiv f \pmod{T^{n+1} F \llbracket T \rrbracket}. \end{equation*}

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Deduce that $f \in R \llbracket T \rrbracket\text{.}$

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Hint.

$F$$f$$T^{n+1}F \llbracket T \rrbracket$$x_1,\dots,x_n,y_1,\dots,y_n\text{.}$

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2.

Prove that there is a unique (up to unique natural isomorphism) functor $\Lambda$ from rings to rings with the following properties.

The underlying functor from rings to additive groups takes $R$ to $\Lambda(R) = 1 + T R \llbracket T \rrbracket$ with the usual series multiplication.

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For any ring $R\text{,}$ the multiplication map $\ast$ on $\Lambda(R)$ satisfies
\begin{equation*} (1-aT)^{-1} \ast (1-bT)^{-1} = (1-abT)^{-1} \qquad (a,b \in R). \end{equation*}

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The ring $\Lambda(R)$ is (a form of) the ring of big Witt vectors with coefficients in $R\text{.}$

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3.

Let $X_1,X_2$ be two varieties over $\FF_q\text{.}$ Prove that in $\Lambda(\ZZ)\text{,}$ we have

\begin{equation*} Z(X_1 \times_{\FF_q} X_2, T) = Z(X_1, T) \ast Z(X_2, T). \end{equation*}

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4.

Let $K$ be a field of characteristic 0. Let $P(x) \in K[x]$ be a monic polynomial of degree $2g+1$ with no repeated roots.

Let $X$ be the affine scheme $\Spec K[x,y]/(y^2-P(x))\text{.}$ Prove that $\Omega^1_{X/K}$ is freely generated by $dx/y\text{.}$

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Prove that $H^1_{\dR}(X)$ admits the basis
\begin{equation*} x^i \frac{dx}{y} \qquad (i=0,\dots,2g-1). \end{equation*}

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Let $Y$ be the affine scheme $\Spec K[x,y,z]/(y^2-P(x),yz-1)\text{.}$ Prove that $H^1_{\dR}(Y)$ admits the basis
\begin{equation*} x^i \frac{dx}{y}, \qquad (i=0,\dots,2g-1); \qquad x^i \frac{dx}{y^2} \qquad (i=0,\dots,2g). \end{equation*}

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Hint.

$dx/y$$\Omega^1_{X/K}\text{;}$$y=0$$y \neq 0$$d \geq 2g\text{,}$$Q(x)\,dx/y$$\deg(Q) = d\text{.}$

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5.

Let $p>2$ be a prime. Let $\overline{P} \in \FF_p[x]$ be a monic polynomial of degree $2g+1$ with no repeated roots.

Put $\overline{X} = \Spec \FF_p[x,y]/(y^2 - \overline{P}(x))\text{.}$ Prove that $H^1_{\MW}(\overline{X})$ admits the basis
\begin{equation*} x^i \frac{dx}{y} \qquad (i=0,\dots,2g-1). \end{equation*}

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Put $\overline{Y} = \Spec \FF_p[x,y,z]/(y^2-\overline{P}(x),yz-1)\text{.}$ Prove that $H^1_{\MW}(\overline{Y})$ admits the basis
\begin{equation*} x^i \frac{dx}{y}, \qquad (i=0,\dots,2g-1); \qquad x^i \frac{dx}{y^2} \qquad (i=0,\dots,2g). \end{equation*}

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Section 19.6 Supplementary exercises

These exercises were not assigned during the course, but were added subsequently.

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Exercises Exercises

1.

Using the Weil conjectures for curves, show that a curve $X$ of genus $1$ over a finite field $k$ cannot satisfy $X(k) = \emptyset\text{.}$

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2.

Using Tate’s theorem, show that the zeta function of a curve $C$ over a finite field $\FF_q$ is uniquely determined by the sequence

\begin{equation*} \#\Jac(C)(\FF_q), \#\Jac(C)(\FF_{q^2}), \#\Jac(C)(\FF_{q^3}), \dots. \end{equation*}

Optional (and harder): use the Weil conjectures to show that $O(g)$ terms suffice, where $g$ is the genus of $C\text{.}$ See [74].

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3.

Let $X$ be a geometrically irreducible variety over a finite field $\FF_q\text{.}$ Using the Weil conjectures, show that there exists an integer $N$ such that $X(\FF_{q^n}) \neq \emptyset$ for all $n \geq N\text{.}$

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4. Weil’s lemma.

Let $\ell$ be a prime. Let $P \in \QQ_\ell[T]$ be a polynomial with roots $\alpha_1,\dots,\alpha_d \in \overline{\QQ}_\ell\text{.}$ Show that $P$ is uniquely determined by the function $\ZZ[T] \to \QQ$ given by

\begin{equation*} F \mapsto \left| \prod_{i=1}^d F(\alpha_i) \right|_\ell. \end{equation*}

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5.

Fix a positive integer $g$ and a prime power $q\text{.}$ Using the Weil conjectures, show that the number of polynomials that can occur as $P_1(T)$ for some abelian variety of dimension $g$ over $\FF_q$ is bounded.

Hint.

$g$$q\text{.}$

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6.

Let $X = \Spec(\overline{A})$ be an affine scheme of finite type over $\FF_q\text{.}$ Prove that $\#X(\FF_q) = 0$ if and only if the ideal in $\overline{A}$ generated by all elements of the form $f^q - f$ for $f\in \overline{A}$ is the unit ideal.

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7.

Put $X = \Spec(k)$ for some field $k$ and fix an algebraic closure $\overline{k}$ of $k\text{.}$ Show that the profinite fundamental group $\pi_1(X,\Spec(\overline{k}))\text{,}$ as defined in Definition 15.1.5, is canonically isomorphic to the absolute Galois group $G_k = \Gal(\overline{k}/k)\text{.}$

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8.

Let $G$ be a profinite topological group. Prove that any homomorphism $G \to \GL(r,\overline{\QQ}_\ell)$ has image contained in $\GL(r,E)$ for some finite extension $E/\QQ_\ell\text{.}$

Hint.

[70]

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9.

Let $p$ be a prime congruent to 3 modulo 4. Show that the elliptic curves

\begin{equation*} y^2 = x^3 - x, \qquad y^2 = x^3 + 4x \end{equation*}

over $\FF_p$ are isogenous, but one has full rational 2-torsion while the other does not. This provides an example of the phenomenon described in Remark 6.0.12.

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10.

Let $L/K$ be an extension of fields. Let $V_1, V_2$ be two $K$-vector space of the same dimension. Let $W$ be a subspace of $\Hom_K(V_1, V_2)\text{.}$ Prove that $W$ contains an element of full rank if and only if $W \otimes_K L \subseteq \Hom_L(V_1 \otimes_K L, V_2 \otimes_K L)$ does.

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