Section 2 Abelian varieties
What about generalising this method to abelian varieties?
For \(p\) odd Coates et. al. (ppav with \(p\)-cyclic isogenies and local constraints)
For \(p = 2\text{.}\)
Recall let \(X,Y/K\) be abelian varieties over a number field and suppose that \(\Psi\colon X\to Y\) is an isogeny, then \(\Psi^\vee \colon Y^\vee \to X^\vee\) its dual. Then
on the other hand we showed that if \(\Psi \Psi^\vee = \lb p \rb\) then
note that in this case \(\deg \psi = p^{\dim (X)}\text{.}\)
To be able to use the same method we need to compute the RHS of (2.1).
For \(E\) since \(E\simeq E^\vee\) and \(|\Sha_0(E)| = \square \text{,}\) this only meant computing
First consider a ppav \(X/K\) s.t.
-
Can we compute
\begin{equation} \frac{\prod_v c(X/K_v)}{\prod_v c(Y/K_v)} \frac{\Omega_X}{\Omega_Y}\text{?}\tag{2.3} \end{equation}Leads us to Jacobians of hyperelliptic curves of genus \(g\)
-
Can we compute
\begin{equation} \frac{|\Sha_0(X) \lb p^\infty\rb |}{|\Sha_0(Y) \lb p^\infty\rb |}\text{?}\tag{2.4} \end{equation}Leads us to Jacobians of hyperelliptic curves of genus \(g\)
-
Need an isogeny \(\Psi\) of degree \(2^g\) s.t.
\begin{equation*} \Psi \colon J \to J' \end{equation*}i.e. the codomain must be a Jacobian of a hyperelliptic curve otherwise we cannot compute 1. or 2.
To satisfy 1., 2. and 3. we take \(g = 2\) because of the following:
Theorem 2.1. González, Josep, Jordi Guardia, and Victor Rotger. Abelian surfaces of GL2-type as Jacobians of curves. arXiv preprint math/0409352 (2004).
Let \(A/K\) be a principally polarized abelian surface defined over a number field. Then \(A\) is one of the following types
-
\begin{equation*} A/K \simeq_K J(C) \end{equation*}where \(C/K\) is a smooth genus 2 curve.
-
\begin{equation*} A/K \simeq_K C_1 \times C_2 \end{equation*}where \(C_1,C_2/K\) are elliptic curves defined over \(K\text{.}\)
-
\begin{equation*} A/K \simeq_K \Res_{F/K} C \end{equation*}where \(\Res_{F/K} C\) is the Weil restriction of an elliptic curve defined over a quadratic extension \(F/K\text{.}\)
Remark 2.2.
The parity of the rank of \(A/K\) in the last two cases can be computed from that of the underlying elliptic curves.
We will concentrate on \(A\simeq_K J(C)\text{,}\)
for \(\deg (f ) = 6\text{.}\)
The generalisation of a 2-isogeny is called a Richelot isogeny.
Plan:
- Review of hyperelliptic curves and their Jacobians.
- Richelot isogeny
- Compute contribution of the real places
- Compute Tamagawa numbers/local root numbers
- Compute \(|\Sha_0(J) \lb 2^\infty \rb|\) up to squares
- Find and prove the right error term
Subsection 2.1 Review of hyperelliptic curves and Jacobians
See Stoll's notes.
By a hyperelliptic curve \(C\) over a number field \(K\) given my
of genus \(g\) where \(f(x) \in K\lb x\rb\) of degree \(2g+1\) or \(2g+2\) with no multiple roots, we mean the pair of affine patches
glued together along the maps
We refer to as the points at \(\infty\) (i.e. \(C \smallsetminus U_x\)) the points with \(t = 0\) on \(U_t\text{.}\)
Explicitly denote by \(c\) the leading term of \(f(x)\text{.}\)
If \(f(x)\) is of degree \(2g+1\) then
we denote \(P_\infty = (0,1)\) the only point at infinity with \(t=0\text{.}\)
Otherwise if \(f(x)\) is of degree \(2g+2\) then
we denote \(P_\infty^\pm = (0,\pm \sqrt{c})\) the two points on \(U_t\) with \(t = 0\text{.}\)
Divisors and the picard group.
Let \(G_K\) be the absolute galois group of \(K\text{,}\) recall that \(G_K\) acts on
via its action on coordinates.
Definition 2.3.
A divisor \(D\) on \(C\) is a formal sum
where \(n_P \in \ZZ\) and \(n_P = 0\) for all but finitely many \(P \in C(K^\sep)\text{.}\) The integer \(n_P\) is called the multiplicity of \(P\) in \(D\) and \(\deg(D) = \sum_{P} n_P\) is the degree of \(D\text{.}\)
Divisors on \(C\) are elements of the free abelian group on the set of points \(P\in C(K^\sep)\text{.}\) Denote by \(\Div(C)\) the group of divisors on \(C\text{.}\)
Definition 2.4.
A divisor
for some Galois extension \(F|K\text{.}\) We say it is \(K\)-rational, or defined over \(K\) if
Example 2.5.
then
is a \(K\)-rational divisor.
Definition 2.6.
Let \(f\) be a non-zero rational function on \(C\text{.}\) Define
where the multiplicity of \(P \) in \(\lb f\rb\) is given by the order of vanishing of \(f \) at \(P\text{.}\) These divisors are called principal divisors, the group of such is denote \(\Princ(P)\text{.}\) Note that these are all of degree 0.
Definition 2.7.
The picard group of \(C\) is defined to be
Note that this inherits a notion of degree from \(\Div(C)\text{.}\)
Theorem 2.8.
Let \(C\) be a smooth, projective, absolutely irreducible curve of genus \(g\) over some field \(K\text{.}\) Then there exists an abelian variety \(J\) of dimension \(g\) over \(K\) s.t. for each field
Definition 2.9.
\(J\) is called the Jacobian variety of \(C\text{.}\)
Remark 2.10.
\(J\) is a projective variety (abelian), thus it can be embedded in some projective space \(\PP^N\) over \(K\text{.}\) One can show that
always works for hyperelliptic curves.
This is too large to work with an explicit model for \(J\) instead we will work with the curve \(C\text{.}\)
Jacobians of genus 2 curves.
Let \(C\) be a hyperelliptic curve of genus 2 defined over \(K\text{.}\)
with \(f(x) \in K\lb x \rb\) of degree 6.
Points on \(C(\overline K)\) and \(J(\overline K)\text{:}\)
A point \(D\) on \(J(\overline K)\) is given by a divisor on \(C\) of the form
for some \(P,Q \in C(\overline K)\text{.}\) For \(D\) to be defined over \(K\) either \(P,Q \in C(K)\) or \(P = Q^\sigma\) for \(\sigma \in \Gal F K\) where \(\lb F : K \rb = 2\text{.}\)
Remark 2.11.
If \(P = (x,y)\) and \(P' = (x,-y)\) then
is zero in \(J(\overline K)\text{.}\)
Addition:
Choose 4 points \(P,P',Q,Q'\in C(\overline K)\) (in general position to make it easier).
We can find a cubic polynomial \(y = p(x)\) through the four points. It also intersects at two additional points \(S,S'\) so that
hence
where \(\lb R , R'\rb = -\lb S, S'\rb\text{.}\) Where negation is taking negative of all \(y\)-coordinates.
So what is 2-torsion?
Lemma 2.13.
Each non-zero element of \(J(\overline K) \lb 2 \rb\) may be uniquely represented by the following pairs of points on \(C(\overline K)\text{,}\) let \(x_1, \ldots, x_6\) be the roots of \(f(x)\) then
Remark 2.14.
For the Richelot isogeny \(\phi\text{:}\)
where \(\phi^\vee \circ \phi = \lb 2 \rb\) and \(\Gamma\) is a correspondence.
Subsection 2.2 Richelot isogenies and the Richelot construction
Richelot isogenies are defined for Jacobians of genus 2 curves, they split multiplication by 2. Their codomain is the Jacobian of a curve, a model of which is explicitly given by the Richelot construction.
Definition 2.15. The Richelot operator.
Given two polynomials \(P(x), Q(x)\in K\lb x\rb\) of degree at most 2 we define the Richelot operator \(\lb -, - \rb\) by
Definition 2.16. Richelot polynomials.
We say that a polynomial \(G(x) \in K \lb x \rb\) of degree 5 or 6 is a Richelot polynomial over \(K\) if we can fix a factorisation
where each \(G_i\) is of degree at most 2, defined over \(\overline K\) and defined over \(K\) as a set.
Write
for its factorisation over \(\overline K\) and define
Definition 2.17. Richelot dual polynomials.
To a Richelot polynomial \(G(x)\) with a fixed factorisation
such that \(\Delta_G \ne 0\text{.}\) We associate its Richelot dual polynomial \(F(x)\) given by
where we take indices mod 3. Write \(F_i(x) = f_i(x-A_i)(x-B_i)\)
\(\Delta_G\) may not be defined over \(K\) but \(\Delta_G^2\) is.
Definition 2.18. Richelot (dual) curves.
We say that a hyperelliptic curve \(C/K\) of genus 2 is a Richelot curve over \(K\) if it is given by \(y^2 = G(x)\) together with the factorisation
as a Richelot polynomial over \(K\) such that \(\Delta_G \ne 0\text{.}\)
To a Richelot curve \(C/K\) we associate its Richelot dual curve \(\widehat C\) given by
where \(F(x)\) is the Richelot dual polynomial of \(G(x)\) with respect to the given factorisation.
Remark 2.19.
Let \(G(x) \in K \lb x \rb\) be a polynomial of degree 5 or 6. Denote by \(K_G\) its splitting field. Then the conditions for \(G(x) \) to be a Richelot polynomial can be rephrased as
Richelot isogenies.
Definition 2.20. Richelot isogenies.
Let \(C/ K\) be a Richelot curve with fixed factorisation
Let \(J\) be its Jacobian, consider the 2-torsion points of \(J(\overline K)\) defined by the quadratic factorisation of \(G(x)\text{.}\)
where \(P_i = (\alpha_i, 0)\text{,}\) \(Q_i = (\beta_i, 0)\text{.}\) Then the isogeny over \(K\) for \(J\) whose kernel is \(\{0, D_1, D_2, D_3\}\) is called a Richelot isogeny.
We say that a Jacobian admits a Richelot isogeny over \(K\) if its underlying curve is a Richelot curve \(/K\text{.}\)
Theorem 2.21.
Let \(C/K\) be a Richelot curve with fixed factorisation
Let \(\widehat C /K\) be its Richelot dual curve and let \(\phi\) denote the associated Richelot isogeny on \(J\text{.}\) Then \(\phi\colon J \to \widehat J\) where \(\widehat J\) is the Jacobian of \(\widehat C\) and moreover \(\hat \phi \phi = \lb 2\rb\text{.}\)
Brauer groups Galois cohomology and local invariants (Angus).
Reference Milne's CFT.
Central simple algebras:
We will consider finite dimensional \(k\)-algebra for \(k\) a field.
Definition 2.22.
A \(k\)-algebra \(A\) is central if the center \(Z(A) = k\text{.}\) A \(k\)-algebra is simple if the only two sided ideals are \(A\) and \((0)\text{.}\)
Example 2.23.
The matrix algebra \(M_n(k)\) is central simple for \(k\text{.}\)
Example 2.24.
A quaternion algebra like \(\HH = \RR \{i,j,k\}\) is central simple for \(k\text{.}\)
Example 2.25.
A division algebra is simple.
Definition 2.26.
Two central simple \(k\)-algebras \(A,B\) are similar, if there exists \(m,n \in \ZZ_{\gt 0}\) s.t. \(A \otimes_k M_m(k) \simeq B \otimes_k M_n(k)\text{.}\) Denote this by \(A\sim B\text{.}\)
Definition 2.27. Brauer groups.
The Brauer group of a field \(k\) denoted \(\Br(k)\) is the set of similarity classes of central simple algebras \(\lb A \rb\) with operation
Remark 2.28.
- The class \(\lb M_n(k)\rb\) is the identity for all \(n\text{.}\)
- The operation is well defined.
- Given \(A\) let \(A^\op\) be the algebra with order of multiplication reversed. Then\begin{equation*} A \otimes_k A^\op \xrightarrow \sim \End_k(A) \simeq M_{\dim_k(A)} (k) \end{equation*}\begin{equation*} (a\otimes a') \mapsto (v \mapsto a v a')\text{.} \end{equation*}So\begin{equation*} \lb A \rb\inv = \lb A^\op \rb\text{.} \end{equation*}
Galois cohomology:
Theorem 2.29. Noether-Skolem.
Let \(A, B\) be central simple \(k\)-algebras and \(f,g\colon A \to B\) a \(k\)-algebra morphism. Then there exists
such that
Let \(A\) be a central simple \(k\)-algebra with maximal subfield \(L/k\text{.}\)
Let \(\sigma \in \absgal k\text{,}\) it induces a map
comparing this to the identity Noether-Skolem gives an element
defined up to multiplication by \(L^\times\text{.}\)
Given another \(\tau \in \absgal k\) I have
thus there exists
s.t.
this gives a map
Theorem 2.30.
This descends to
Some special \(k\text{.}\)
Theorem 2.31. Wedderburn.
Every central simple \(k\)-algebra is isomorphic to \(M_n(D)\) for \(D\) a division \(k\)-algebra.
Proposition 2.32.
If \(k = \overline k\) then any division \(k\)-algebra \(D\) is isomorphic to \(k\text{.}\) Thus \(\Br(k) = 0\text{.}\)
Theorem 2.33. Wedderburn.
Every finite division ring is a field. So if \(k\) is a finite field then \(\Br(k) = 0\text{.}\)
Theorem 2.34. Frobenius.
Every central division \(\RR\)-algebra is isomorphic to either \(\RR\) or \(\HH\text{.}\) Thus \(\Br(\RR) \simeq \ZZ/2\text{.}\)
Let \(k\) be a non-archimidean local field with valuation
for a central division algebra \(D\) there exists \(n \in \ZZ\) s.t.
Consider a maximal unramified subfield
with \(\sigma\in \Gal LK\) lifting frobenius.
Noether-Skolem gives \(\alpha \in D^\times\) s.t.
up to \(L^\times\text{.}\)
If we take \(\alpha ' = c\alpha \) for \(c\in L^\times\) we can compute
We get a map
Theorem 2.35.
This descends to an isomorphism
If \(F\) is a number field with a place \(v\in |F|\) get a map
Global CFT gives an exact seq
Root numbers of elliptic curves (Ricky).
Based on Rohrlich's article elliptic curves and the Weil-Deligne group
\(K\) non-archimidean local field, \(\overline K\) is its separable closure.
\(\Phi\) some lift of \(\phi\) in \(\absgal K\text{.}\)
\(W(\overline K / K) = \) Weil group, the preimage of \(\langle \phi\rangle\) in \(\absgal k\) under \(G_K \twoheadrightarrow G_k\text{.}\)
We consider \(\sigma \colon W( \overline K/K) \to \GL(V)\text{,}\) representations over \(V/\CC\) (always cts.)
Say \(\sigma\) is of Galois type if it factors through a finite quotient.
Another source of examples is
given by
where
and \(\omega(\Phi) = q\inv\text{.}\)
Fact, all irreducible \(\sigma \cong \rho \otimes \omega^s\) for some \(s \in \CC\) and \(\rho\) of Galois type.
Definition 2.36. The Weil-Deligne group.
The Weil-Deligne group is
where \(W\) acts on \(\CC\) via \(\omega\)
Upshot: Representations \(\sigma'\) of \(W'\) are the same as \((\sigma, N)\) where
a representation and \(N\) is a nilpotent linear operator on \(V\text{.}\) Satisfying
One motivation for studying those is a general construction of Grothendieck and Deligne which turn an \(l\)-adic representation of \(G_K\) into a representation of \(W'\) (given \(i\colon \QQ_l \hookrightarrow \CC\)).
Example 2.37.
with action of \(W'\) given by
check relation \(\sigma N \sigma\inv = \omega N\text{.}\)
We want to define \(\epsilon \)-factors for representations of \(W'\text{.}\) We need two choices:
an additive character of \(K\text{.}\) And
a Haar measure on \(K\text{.}\)
Then
where
and \(\epsilon (\sigma , \psi , \diff x)\) is defined by the following proposition.
Proposition 2.38. Deligne-Langlands.
There exists a unique function \(\epsilon (\sigma ,\psi \diff x)\) satisfying
- \(\epsilon (*, \psi , \diff x)\) is multiplicative in short exact sequences.
- If \(L/K\) is finite then\begin{equation*} \epsilon (\Ind_{L/K} \rho , \psi , \diff x) = \epsilon (\rho , \psi \circ \trace_{L/K}, \diff x_L) \cdot \left( \epsilon (\Ind_{L/K} 1_L, \psi , \diff x) / \epsilon (1_L, \psi \circ \trace_{L/K}, \diff x_L)\right)^{\dim \rho } \end{equation*}
- For \(\chi \) a character\begin{equation*} \epsilon (, \chi , \psi , \diff x) \end{equation*}agrees with the ones defined in Tate's thesis. They're both given by an integral formula.
Definition 2.39. Root numbers.
The root number of \(\sigma '\) is defined to be
For \(E/K\) an elliptic curve we have a representation on \(V_l^*\) (\(l \ne p\)).
Using the Grothendieck-Deligne construction, let \(\sigma _{E/K}\) be a representation of \(W'\) it has the following property
- \(E\) pot. good reduction then\begin{equation*} N_{E/K} = 0 \end{equation*}and \(\sigma_{E/K}\) is semisimple. \(E\) has good reduction iff \(\sigma _{E/K}\) is unramified.
- \(E\) has potential multiplicative reduction implies that we can take \(\chi \) a character of \(W\) with \(\chi ^2 = 1\text{,}\) so that\begin{equation*} E^\chi \end{equation*}has split multiplicative reduction. Then\begin{equation*} \sigma _{E/K}' \simeq \chi \omega \inv \otimes \sprep(2) \end{equation*}\(\chi \) is trivial / unramified and non-trivial / ramified according to \(E\) having split / non-split / additive reduction.
- \(\sigma _{E/K}'\) is essentially symplectic. \(W(E/K) = W(\sigma _{E/K}')\) is independent of \(\psi \) and must be \(\pm 1\text{.}\)
Proposition 2.40.
- \(E\) has good reduction implies \(W(E/K) = 1\text{.}\)
- \(E\) potentially multiplicative reduction implies\begin{equation*} W(E/K) = \begin{cases} -1 \amp \text{split}\\ 1 \amp \text{nonsplit}\end{cases}\text{.} \end{equation*}If additive reduction take \(\xi \) quadratic character s.t.\begin{equation*} E^\xi \end{equation*}has split multiplicative reduction and \(W(E/K) = \xi (-1)\text{.}\)
\(\Sha\) (Sachi).
Suppose \(G\) is a finite abelian group with a non-degenerate alternating, bilinear paring
then there exists \(H\) s.t. \(G \cong H\times H\text{.}\)
Nondegeneracy is the property that: If \(\Gamma (v,w) = 0\) for all \(w \in G\) then \(w= 0\text{.}\)
Alternating: For all \(v \in G\text{,}\) \(\Gamma (v,v ) = 0\text{.}\) (this implies skew-symmetry).
Analogous theorem:
Symplectic space if \(V\) a vector space with non-degenerate alternating bilinear pairing, \(\omega \) has a decomposition.
where \(W\) is Lagrangian.
Proof is via induction on the dimension of \(V\text{.}\) Fix \(v\in V\text{.}\) \(\exists W\) s.t. \(\omega (v,w) = 1\text{,}\) scalar nondegeneracy.
Define \(W = \{z\in V : \omega (z,w) = 0, \omega (v,z) = 0\}\text{.}\)
so restrict \(\omega \) to \(W\text{,}\) induct.
Proof of the theorem.
Trivial group \(\checkmark\text{.}\)
Reduce to the case of a \(p\)-group, \(G\) a \(p\)-group. Fix \(x\) of maximal order in \(G\text{,}\) \(p^n\text{.}\) There exists \(y\) such that \(\Gamma (x,y) = \frac 1{p^n}\text{.}\) If not then \(\Gamma (p^{n-1}x,y) = 0\) for all \(y \in G\) so this contradicts non-degeneracy. Any \(y\) has maximal order also since
Next we want to show \(\langle x \rangle \cap \langle y \rangle = 0\text{.}\) If \(mx = ny\) for some \(0 \lt m, n \lt p^n\) then
Define
claim:
Proof of claim: If \(g \in G\)
so
here we used alternating.
Then \(\Gamma \) restricts to a non-degenerate alternating bilinear pairing on \(H\text{.}\)
Remark 2.41.
For a PPAV we do not always have an alternating pairing, sometimes just skew-symmetric, or nothing! So Sha can be square, twice a square, or arbitrary. See Poonen-Stoll, Stein?
Complete 2-descent (Oana).
Let
http://www.lmfdb.org/EllipticCurve/Q/800/d/3, then
so the bad primes are \(2,5\text{.}\)
\(\#\tilde E(\FF_3) = 4\text{.}\)
so
\(S = \{2,5, \infty \} \subseteq M_\QQ\text{.}\)
a complete set of coset representatives is
which has 8 elements. Consider
does the system
have a solution for pairs \((b_1, b_2) \in \QQ(S,2)^2\) and \(z_1, z_2, z_3 \in \QQ\text{?}\)
If \(b_1 \lt 0, b_2 \gt 0\) or \(b_1 \gt 0 , b_2 \lt 0\) then we have no solution.
| \(b_1\) | \(b_2 \) | reason/point? |
| 1 | 1 | point 0 |
| 1 | 2 | |
| 1 | 5 | |
| 5 | 2 | point (0,5) |
| -1 | -1 | point (-4,6) |
| -5 | -2 | point (0,5) + (-4,6) |
Reason if \(\legendre ap = -1\) and \(x^2 = ay^2 \pmod p\) then
then
If \(5\nmid b_1\) and \(\legendre{b_2} 5 = -1\) then
we have \(z_3 \in 5\ZZ_3 \cap \QQ\)
We reverse engineer \((-4,6) \in E(\QQ)\text{.}\)
Weil-Châtelet groups (Aash, Asra).
I have an elliptic curve \(E/K\text{,}\) then \(C/K\) a smooth curve is a PHS if
Such that \(\mu \) is defined over \(K\) and \((P+Q) + p = P+(Q+p)\) and for all \(p,q \in C(\overline K)\) there exists a unique \(P\in E(\overline K)\) s.t. \(\mu (P,p) = q\text{.}\)
We say two PHS \(C,C'\) are equivalent if
which respects the action of \(E\text{.}\)
\(\forall P \in E, p\in C\)
\(\operatorname{WC}(E)\) is set of the equivalence classes of PHS's.
Proposition 2.43. Weil.
Let \(H_1, H_2\) be homogeneous spaces for an algebraic group \(G/K\text{.}\) There exists \(H\) a PHS over \(K\) and
where \(P, Q \in Q\text{,}\) \(p \in H_1, q \in H_2\) this \(H\) is unique up to PHS isomorphism. If \(\mathcal H_1, \mathcal H_2\) are the classes of \(H_1, H_2\) we call \(\mathcal H_1 + \mathcal H_2\) the class of \(H\) (above). This defines a group structure.
- Well defined binary operation
- Identity: call class of \(G\text{,}\) \(\mathcal H_0\text{.}\)
for any \(\mathcal H\text{.}\) Inverse: Say \(H\) is a PHS, consider \(H^-\)
\(P = v(a,b) \in E\) s.t. \(P+ b = a\text{.}\) Associativity: \(H_1, H_2, H_3\)
We showed
Missed ????????
Take \(a \in \Sha(A/K)\) then \(a\) can be represented by a locally trivial PHS \(X\) over \(K\text{.}\) Let \(K^\sep(X)\) be the function field of \(X\otimes_K K^\sep\text{.}\) Have an exact sequence
which yields
the last 0 is as \(H^3(G_K, (K^\sep)^\times) = 0\) as \(X\) is locally trivial (c.f. Mlne Arithmetic duality theory rmk. 6.11) we have
On the other hand from the exact sequence
we have
now over \(K^\sep\text{,}\) \(A\otimes_K K^\sep \simeq X\otimes_K K^\sep\) hence
hence one gets a map
Fact 2.44.
For Jacobians of curves if the principal polarization on \(J\) is given by a rational divisor then \(\pair \cdot \cdot\) is alternating, hence \(|\Sha_0(A/K)| = \square\) otherwise \(|\Sha_0(A/K)| = 2 \square\text{.}\)
Noted by Poonen and Stoll.
Theorem 2.45.
\(C\) is deficient at an odd number of place iff
Definition 2.46. Deficient places.
We say that \(C\) is deficient at a place \(v\) if \(C\) doesn't have a \(K_v\) rational divisor of degree \(g-1\text{.}\)
Hence for genus \(g\) curves this says that \(C\) has no \(K_v\) rational divisor of degree \(1\text{.}\) Equivalently \(C\) has no \(K_v\)-rational point over any odd degree extension of \(K_v\text{.}\)
E.g. if \(K_v = \RR\) we have \(C\) deficient iff \(C(\RR) \ne \emptyset\text{.}\)
Here \(c \gt 0\) and \(C(\RR) \ne \emptyset\) and \(C\) is not deficient over \(\RR\text{.}\)
Alternatively \(c \lt 0\) and \(C(\RR) = \emptyset\) and \(C\) is deficient over \(\RR\text{.}\)
Infinite places.
Definition 2.48.
Let \(J/K\) be a jacobian admitting a Richelot isogeny \(\phi \) over \(K\) for a place of \(K\) such that \(v | \infty \text{,}\) we denote \(\phi _v\) the map induced by \(\phi \) on \(J(K_v)\) and define
the restriction of \(\phi _v\) to the identity component.
Lemma 2.49.
where \(n (J(K_v))\) denotes the number of connected components of \(J(K_v)\text{.}\)
Proof.
Same as the elliptic curve case.
Case \(K_v = \CC\) here \(n(J(\CC)) = 1 = n(J'(\CC))\) and \(|\ker \varphi | = 4\)
Proposition 2.50.
Proposition 2.51.
A divisor \(D_i = \lb P_i, Q_i \rb \in \ker (\phi )\) is in \(\ker \varphi \) iff the points \(P_i, Q_i\) satisfy either
- \(P_i = \overline Q_i\text{,}\) or
- \(P_i\) and \(Q_i\) lie on the same connected component of \(C(\RR)\text{.}\)
\(D_1= \lb (r_1, 0), (r_2,0)\rb\) with \(r_1,r_2\) the smallest roots. Then \(D_1\in \ker \varphi\text{.}\)
\(D_2= \lb (r_1, 0), (r_3,0) \rb \) with \(r_3\) the next smallest root. Then \(D_2\not\in \ker \varphi\text{.}\)
Missed
Proposition 2.54.
The number of real roots of \(F(x)\) (hence \(n( J(K_v))\)) is given as follows (addition modulo 3):
- If \(\delta _i\in \RR \) and \(\delta _{i+1} , \delta _{i+2} \not\in \RR\text{,}\) i.e. \(\delta _{i+1} = \overline \delta _{i+2}\) then\begin{equation*} \delta _i' \in \RR,\,\delta _{i+1}', \delta _{i+2}' \not \in \RR \end{equation*}with\begin{equation*} \delta _{i+1}' = \overline \delta _{i+2}'\text{.} \end{equation*}
- If \(\delta _i, \delta _{i+1}\in \RR \) then \(\delta _{i+2}'\in \RR\text{,}\) and \(\delta _{i+2}' \lt 0\) iff \(k_{i, i+1} \lt 0\text{.}\)
Proof.
Clear since
for \(i = 1,2,3\text{.}\)
Remark 2.55.
\(m_G'\) follows from the signs of \(\delta _1', \delta _2', \delta _3'\) and the leading term of \(F(x)\text{.}\)
Example 2.56.
Let \(G_1(x) = x^2 - 16, G_2(x) + x^2 + x + \frac{17}4, G_3 = x^2 - 2x + 9\text{.}\) We have \(\delta _1=64, \delta _2 = -16, \delta _3 = -32\text{.}\) \(C\) has one real connected component hence \(n(J(\RR)) = 1\) and \(m_v = 1\text{.}\)
Now
so \(|\ker \phi | = 4\text{.}\)
Also \(\delta _{1}' ,\delta _{2}', \delta _{3}'\in \RR\) all \(k_{i,j} \gt 0\) so \(\delta _{1}' ,\delta _{2}', \delta _{3}'\gt 0\) so that \(C'\) has \(3\) connected components and \(n(J'(\RR)) = 4\) and \(m_v' = 1\text{.}\)
Tamagawa numbers (\(v \nmid \infty \)).
We need to compute \(\frac{c_v(J)}{c_v(J')}\) (we won't at \(v | 2\)).
Recall that for an abelian variety \(A/K\) over a number field
Lemma 2.57.
Let \(S\) be a finite set of primes of \(K\) containing archimidean places and bad reduction places. For each place \(v\not\in S\text{,}\) denote \(\widetilde A_v\) the abelian variety over the residue field \(\FF_{q_v}\) where \(q_v = N_{{K_v}/\QQ_v} (v)\text{.}\) Set \(d = \dim A = \dim \widetilde A_{v}\text{.}\) Let \(\omega \ne 0 \) be a choice of exterior differential form of degree \(d\) on \(A\) defined over \(K\) and for \(v\nmid \infty \text{.}\) Consider \(|\omega |_v \mu _v^d\) which determines a Haar measure on \(A(K_v)\text{.}\) Then
where \(\omega _0\) is a choice of \(v\)-regular \(d\)-form with \((\widetilde \omega _0)_v \ne 0\) and if \(A\) had bad reduction at \(v\) then \(\widetilde A_v (\FF_{q_v})|\) is the number of \(\FF_{q_v}\) points on the special fibre \(\widetilde A_v\) of Neron's minimal model.
Sketch of proof.
How to compute \(c_v\text{?}\) Need to compute \(|\widetilde A_v(\FF_{q_v})|\text{.}\)
Example 2.58.
Consider an elliptic curve \(E\text{.}\) Recall that by Hensel's lemma, \(E_0(K_v) \twoheadrightarrow \widetilde E_{ns}(\FF_{q_v})\) let
be the minimal Weierstraß equation for \(E\text{.}\) Let \(\tilde f(x,y) \) be the reduced polynomial mod \(\pi _v\text{.}\) and \(\tilde P(\tilde \alpha , \tilde \beta ) \in \widetilde E_{ns}(\FF_{q_v})\) a point. Since \(P\) is non-singular either
say the latter, then choose any \(x_0 \in \ints_{K_v}\) with \(x_0 \equiv \tilde \alpha \pmod{\pi _v}\) then \(f(x_0, y) = 0\) has \(\tilde f(x_0, \tilde \beta ) = 0\) as \(\beta \) is a simple root. By Hensel's lemma there exists \(y_0 \in \ints_{K_v}\) such that \(\tilde y_0 = 0\) and \(f(x_0,y_0) = 0\text{.}\) So \(P=(x_0, y_0) \in E_0(K)\) reduces to \(\widetilde P\text{.}\)
For non-singular points get points over \(\ints_{K_v}\text{.}\)
Example 2.59.
Missed more sorry
Remark 2.60.
We are interested in “good” models, i.e. we require that
Our model \(\mathcal E /\ZZ\colon y^2=(x+1)(x-p^2)(x+p^2)\) is proper since \(\mathcal E \subseteq \PP^2_{\ZZ_p}\) so that \(\mathcal E (\ZZ_p) = E(\QQ_p)\) but it is singular since its special fibre is.
We need to manipulate \(\mathcal E/\ZZ_p \colon y^2 = (x+1)(x-p^2)(x+p^2)\) s.t.
- \(\mathcal E\) is a model of \(\ZZ_p\text{.}\)
- The generic fibre is \(E/\QQ_p\)
- Only non-singular points of its special fibre can be lifted to points over \(\QQ_p\) on \(E\)
To satisfy 1 and 2, we can do change of variables of the form
We will use only \(y=y_1p\) for now.
The collection of these charts (together with their counterpart at infinity) give a regular model \(\mathcal E\) of \(E/\QQ_p\text{.}\)
So we have four components, all \(\PP^1\) meeting in a square. (There are still singularities on \(\mathcal E\) at intersection points in the special fibre, but they are regular singularities, i.e. the local ring at these points is regular, i.e. we have \(\ideal m_P/ \ideal m_P^2\) dimension 2).
Example 2.62.
Let \(a\in \ZZ_p\text{,}\) \(E \colon y^2 = x^3 + a\text{.}\) \(E\) might be singular at \(P = (0,0)\text{,}\) if \(a\equiv 0\pmod p\) then we degenerate to a cusp. The maximal ideal \(\ideal m_P\) of the local ring is generated by \(x,y,p\text{.}\) If \(a\not \equiv 0 \pmod {p^2}\) then \(v(a) = 1\) and \(p \in a\ZZ_p\text{.}\) But \(a = y^2 - x^3\) so, \(p \in (y^2 - x^3)\ZZ_p \subseteq \ideal m_P^2\text{.}\) So \(x,y\) generate \(\ideal m_P/\ideal m_P^2\) and \(P\) is regular. If \(a \equiv 0 \pmod{p^2}\) then \(\ideal m_P/\ideal m_P^2\) cannot be generated by fewer than 3-elements so \(P\) is not regular.
Proposition 2.63.
Let \(\mathcal C /\ZZ_p\) be an arithmetic surface and \(C/\QQ_p\) be the generic fibre of \(\mathcal C\text{.}\)
- If \(\mathcal C\) is proper then \(C(\QQ_p) = \mathcal C(\ZZ_p)\text{.}\)
- If \(\mathcal C\) is regular and proper then\begin{equation*} C(\QQ_p) = \mathcal C(\ZZ_p) =\mathcal C^0(\ZZ_p) \end{equation*}
where \(\mathcal C^0 \subseteq \mathcal C = \mathcal C \smallsetminus\) singular points.
Remark 2.64.
The smooth part of a proper regular arithmetic surface is large enough to contain all of the rational points on the generic fibre.
Definition 2.65. Neron models.
The Neron model of \(E/K\) is an arithmetic surface \(\mathcal E/K\) whose generic fibre is the given elliptic curve. It is such that every point of \(E\) gives a point of \(\mathcal E\) and such that the group law on \(E\) extends to make \(\mathcal E\) into a group (as a scheme over \(R\)).
Remark 2.66.
Neron models are smooth \(R\)-schemes i.e. for every point \(p\in \Spec (R)\) the fibre is a non-singular variety. However it might have several components and may not be complete. So in general \(\mathcal E\) will not be proper over \(R\text{.}\)
Theorem 2.67.
Let \(E/K\) be an elliptic curve, \(\mathcal C/R\) a proper minimal regular model for \(E/K\) and let \(\mathcal E/R\) be the largest subscheme pf \(\mathcal C/R\) which is smooth over \(R\text{.}\) Then \(\mathcal E/R\) is a Neron model for \(E/K\text{.}\)
Recall we need to compute \(|E(\QQ_p)/E_0(\QQ_p)|\) and we considered the example
Write \(\mathcal E^0\) for \(\mathcal E \smallsetminus \{\text{singularities in special fibre}\}\) then
We saw that the Neron model of \(E/\QQ_p\) can be obtained from \(\mathcal E/\ZZ_p\) be removing the singularities in the special fibres.
Proposition 2.68.
\(\mathcal E\) is the Neron model of \(E/\QQ_p\text{,}\) and \(\mathcal E^0/\ZZ_p\) denotes the identity component of \(\mathcal E/\ZZ_p\text{.}\) \(\overline{\mathcal E} /\FF_p\) denotes the special fibre of \(\mathcal E\text{.}\) \(\overline{\mathcal E}^0/\FF_p\) is the identity component of \(\overline{\mathcal E}/\FF_p\text{.}\)
In our case the Tamagawa number is
To actually calculate this use Tate's algorithm.
Subsection 2.3 Jacobians of hyperelliptic curves
Let \(A/\QQ_p\) be such a Jacobian. \(A\) admits a Neron model \(\mathcal A/\ZZ_p\text{.}\) The open subscheme whose special fibre is the connected component of the identity \(\widetilde {\mathcal A}^0\text{.}\)
As for an elliptic curve write \(A_0(\QQ_p)\) for the points reducing to \(\widetilde {\mathcal A}^0(\FF_p)\text{,}\) then
is finite and
How to compute \(c_p\text{?}\)
Theorem 2.69.
Let \(C/\QQ_p\) be a smooth proper, geometrically connected curve, let \(\mathcal C/\ZZ_p\) be the minimal regular model for \(C\text{,}\) \(J/\QQ_p\) its jacobian, \(\mathcal A/\ZZ_p\) the Neron model for \(J\text{.}\)
If \(\mathcal J/\QQ_p\) is semistable (ordinary double roots as singularities) then
as a consequence
Where \(M\) is any minor of the incidence matrix \(N_{ij}\) of
\(N_{ij}\) is the size of the intersection of \(\Gamma _i, \Gamma _j\) for the irreducible components \(\Gamma \) of \(\overline{ \mathcal C}/\overline{\FF_p}\text{.}\)
Example 2.70.
In order to compute \(c_p\) for \(J\) need to construct special fibre of minimal regular model for \(C\text{.}\)
Namikawa-Ueno classification of types of semistable reductions of genus 2 curves.
- Good reduction: \(g = 2\)
- One node.
- Two nodes.
- Three nodes.
- One cusp (triple root).
- Two cusps (triple root).
Cassels-Tate pairing (Maria).
Claim 2.71.
\(K\) number field and \(A/K\) a.v. admits a principal polarization \(\phi _D\) given by a rational divisor. Then the Cassels-Tate pairing
is alternating.
Proof.
by sending \(a\in A(K^\sep)\text{,}\) \(\phi _D(a) = \lb D_a- D\rb\text{,}\) where \(D_a= D+a\) is the translate of \(D\) by \(a\text{.}\)
Assume \(D\) is a rational divisor, what we'll prove is that \(\pair a{\phi _D(a)} = 0\) for all \(a \in A\text{.}\) Where
Fix \(a \in \Sha(A,K) \subseteq H^1(G_K, A)\) and let \(X\) be the corresponding PHS of \(A\text{.}\) Then for any \(P \in X(K^\sep)\text{,}\) \(a\) is represented by the cocycle
Denote \(a' = \phi _D(a)\text{.}\) Then \(a'\) is represented by
this lifts to
which is a cocycle:
using \(K\)-rationality of \(D\text{.}\)
Using
we can regard
now use
Big diagram to conclude.
Subsection 2.4 Semistable models of hyperelliptic curves of genus 2
Recall: Can compute Tamagawa numbers of semistable Jacobians of genus 2 curves from the special fibre of their minimal regular models (i.e. there exists a formula for them.
Definition 2.72.
A model is semistable if its special fibre is geometrically reduced and has only ordinary double points as singularities. When such a model exists over \(K\) we say that the curve is semistable over \(K\text{.}\) Or has semistable reduction \(/K\text{.}\)
Example 2.73.
\(p \ge 7\) and
a node
Have 1 genus 1 component meeting 3 genus 0 in a square on the special fibre of minimal regular model
Example 2.75.
\(p \ge 7\) and
two nodes.
Have 7 genus 0 components meeting in a pair of squares with one common line.
Example 2.77.
\(p \ge 7\) and
two nodes.
Have 8 genus 0 components, two non-intersecting lines joined by 3 chains of two \(\PP^1\)s.
Example 2.79.
\(p \ge 7\) and
a cusp.
Have 2 genus 1 components meeting.
Example 2.81.
\(p \ge 7\) and
two cusps.
Have 2 genus 1 components joined by a \(\PP^1\text{.}\)
Places of \(K\) above \(2\).
Here it is very difficult to compute a minimal regular model for \(C\text{,}\) hence can't compute the Tamagawa numbers.
One way around is to use the definition of the local contribution
Proposition 2.83.
Consider the family
such that \(t_i \in \ints_K\text{,}\) \(t_2 \equiv 1 \pmod 2\text{,}\) \(t_3 - \frac 14 \equiv 0 \pmod 2\text{,}\) \(t_4 \equiv - 2 \pmod 8\text{,}\) \(t_5 \equiv 1 \pmod 8\text{.}\) Then \(C \in \mathcal F\) has totally split toric reduction, and assuming that \(G_2(x),G_3(x)\) are both irreducible over \(K_v\) then
So combining \(v | \infty \text{,}\) \(v\nmid 2\infty \text{,}\) \(v | 2\) if \(C \in \mathcal F\) over \(K_v\) for \(v| 2\) and semistable at \(v \nmid 2\infty \) with \(C\) a Richelot curve then we can compute the parity of the \(\rk_2(J)\text{.}\)
Example 2.84.
have \(C/\QQ_p\) for \(p = 3,5,11,13, 17, 97, 1201\) and \(p = 131\) is good reduction for \(C\) but not for \(C'\text{.}\) For \(p =3,17\) have \(c_p=2, m_p=1\text{.}\) For \(p =5, 11, 13, 97, 1201\) have similar with non-split nodes. \(p =131\) have \(c_p = 1,m_p = 1\text{,}\) \(c'_p = 1, m'_p =2\text{.}\) Two e.c.s swapped.
for \(p = 2\)
hence
so \(J\) has even \(\rk_2(J)\text{.}\)