2016-03-09-bsd-part3-abelian-varieties.sagews

Birch and Swinnerton-Dyer for Abelian Varieties over Number Fields

William Stein

Recall, an abelian variety $A$ over a number field $K$ is by definition (a geometrically irreducible) projective variety equipped with a group structure.

There is an $L$-series $L(A,s)$ attached to $A$, which is got by counting points on $A$ modulo primes $p$.

Conjecture: $L(A,s)$ extends to a holomorphic function on $\CC$, which satisfies a functional equation.

BSD Rank Conjecture: $\text{ord}_{s=1} L(A,s) = \text{rank}(A(K))$

BSD Formula: I think this is the BSD formula over a general number field: $$\frac{L^{(r)}(A,1)}{r!} = \frac{\sqrt{|D_K|}\cdot \Omega_A \cdot \prod c_{\mathfrak{p}} \cdot \text{Reg}(A/K) \cdot \#\text{Sha}(A/K)}{\# A(K)_{\rm tor} \cdot \# A^{\vee}(K)_{\rm tor}}$$

Dual abelian variety: Here $A^{\vee}$ is the dual abelian variety of $A$. It is isogenous to $A$, and often isomorphic to $A$, e.g., when $\text{dim}(A)=1$ then $A=A^{\vee}$. If $A \to B$ is an isogeny of abelian varieties with kernel $G$, then there is a dual exact sequence $B^{\vee} \to A^{\vee}$ with kernel the (Cartier) dual of the group scheme $G$. Jacobian of curves $\text{Jac}(X)$ are also isomorphic to their dual if the curve has a rational point.

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**Exercise:** Write down the simplest elliptic curve you can that has $j$-invariant NOT in $\QQ$, then compute everything
Sage (easily) lets you about the BSD formula for that elliptic curve.    What happens?

Exercise: Write down the simplest elliptic curve you can that has $j$-invariant NOT in $\QQ$, then compute everything Sage (easily) lets you about the BSD formula for that elliptic curve. What happens?

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## BSD over $\QQ$ implies BSD over all number fields!


Amazingly, if we somehow know BSD just for abelian varieties over $\QQ$, we would automatically know it over all number fields!

**Theorem (Milne):** If (either) BSD is true for all abelian varieties over $\QQ$ if only if it is true for all
abelian varieties over number fields.

Why?

There is a construction called "restriction of scalars".  Given a variety $V$ over a number field $K$, there exists
a variety $R = \text{Res}_{K/\QQ}(V)$ over $\QQ$ with the property that for all number fields $M$, we have
$$
   R(M) = V(K\otimes M).
$$
In particular, $R(\QQ) = V(K)$.  You can construct $R$ (locally) by replacing the indeterminates in the defining
polynomials of $V$ by $X_1 \omega_1 + \cdots X_n \omega_n$, where $\omega_1,\ldots, \omega_n$ is a basis for $K$ over $\QQ$,
multiplying everything out, and equating coefficients of $\omega_1$, of $\omega_2$, etc.  If $V$ is an abelian
variety, then so is $R$.  Also, $R_K \approx V \times \cdots \times V$, so geometrically $R$ is a "product of $[K:\QQ]$
copies of $V$".

Applying this construction to an abelian variety $A/K$ yields an abelian variety $R/\QQ$, and in a sense "everything" about $R$ is the same as $A$.  For example, $L(A/K,s) = L(R/\QQ,s)$, and $R(\QQ) = A(K)$, and $\text{Sha}(R/\QQ) \cong \text{Sha}(A/K)$, etc., etc.
In particular, BSD (rank or formula) is true for $A/K$ if and only if it is true for $R/\QQ$.

BSD over $\QQ$ implies BSD over all number fields!

Amazingly, if we somehow know BSD just for abelian varieties over $\QQ$, we would automatically know it over all number fields!

Theorem (Milne): If (either) BSD is true for all abelian varieties over $\QQ$ if only if it is true for all abelian varieties over number fields.

Why?

There is a construction called "restriction of scalars". Given a variety $V$ over a number field $K$, there exists a variety $R = \text{Res}_{K/\QQ}(V)$ over $\QQ$ with the property that for all number fields $M$, we have $$R(M) = V(K\otimes M).$$ In particular, $R(\QQ) = V(K)$. You can construct $R$ (locally) by replacing the indeterminates in the defining polynomials of $V$ by $X_1 \omega_1 + \cdots X_n \omega_n$, where $\omega_1,\ldots, \omega_n$ is a basis for $K$ over $\QQ$, multiplying everything out, and equating coefficients of $\omega_1$, of $\omega_2$, etc. If $V$ is an abelian variety, then so is $R$. Also, $R_K \approx V \times \cdots \times V$, so geometrically $R$ is a "product of $[K:\QQ]$ copies of $V$".

Applying this construction to an abelian variety $A/K$ yields an abelian variety $R/\QQ$, and in a sense "everything" about $R$ is the same as $A$. For example, $L(A/K,s) = L(R/\QQ,s)$, and $R(\QQ) = A(K)$, and $\text{Sha}(R/\QQ) \cong \text{Sha}(A/K)$, etc., etc. In particular, BSD (rank or formula) is true for $A/K$ if and only if it is true for $R/\QQ$.

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## Special case: abelian varieties of $\text{GL}_2$-type

Suppose $A/\QQ$ is a simple abelian variety of $\text{GL}_2$-type, i.e., $\text{End}(A/\QQ)$ is an order in a number field
of degree $[A:\QQ]$.

**Theorem (Khare-Wintenberger-Ribet-Serre):** There is some newform $f\in S_2(\Gamma_1(N))$ such that $A$ is isogenous to
the abelian variety $A_f = J_1(N)/I_f J_1(N)$, where $I_f = \text{Ann}_{\mathbf{T}}(f)$.

Special case: abelian varieties of $\text{GL}_2$-type

Suppose $A/\QQ$ is a simple abelian variety of $\text{GL}_2$-type, i.e., $\text{End}(A/\QQ)$ is an order in a number field of degree $[A:\QQ]$.

Theorem (Khare-Wintenberger-Ribet-Serre): There is some newform $f\in S_2(\Gamma_1(N))$ such that $A$ is isogenous to the abelian variety $A_f = J_1(N)/I_f J_1(N)$, where $I_f = \text{Ann}_{\mathbf{T}}(f)$.

X = Newforms(43, names='a')
print X
[f, g] = X

[q - 2*q^2 - 2*q^3 + 2*q^4 - 4*q^5 + O(q^6), q + a1*q^2 - a1*q^3 + (-a1 + 2)*q^5 + O(q^6)]

E = f.abelian_variety(); E

Newform abelian subvariety 43a of dimension 1 of J0(43)

A = g.abelian_variety(); A

Newform abelian subvariety 43b of dimension 2 of J0(43)

E.intersection(A)

(Finite subgroup with invariants [2, 2] over QQ of Newform abelian subvariety 43a of dimension 1 of J0(43), Simple abelian subvariety of dimension 0 of J0(43))

A + E

Abelian variety J0(43) of dimension 3

f.lseries()(1)

0.000000000000000

g.lseries()(1)

0.620539857407845

L = g.lseries()

There is an algorithm (from my Ph.D. thesis) to compute $L(A_f,1)/\Omega_{A_f}$...

# I can't believe I haven't implemented this in Sage yet :-(
A.lseries().rational_part()

Error in lines 2-2 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/lseries.py", line 152, in rational_part raise NotImplementedError NotImplementedError

Let $\Phi_f: H_1(X_0(N),\ZZ,cusps) \to \CC^d$ given by integrating a modular symbol against all the Galois conjugates of $f$. The algorithm is that $$\frac{L(A_f,1)}{\Omega_{A_f}} = [\Phi_f(H_1(X_0(N),\ZZ)^+): \Phi_f(\mathbf{T} \{0,\infty\})]$$ where the index on the right is a lattice index, so neither side need be contained in the other, so the resulting index is a rational number. Define $$[L:L'] := [L+L':L']/[L+L':L].$$

(NOTE: there is a subtle issue involving Manin constants that I'm ignoring.)

MS = g.modular_symbols(sign=0)
# The integral_period_mapping function normalizes things so the image
# H_1(X_0(N),Z)/* is just ZZ^g.
Phi = MS.integral_period_mapping()
# So we just need to compute Phi(T*e), where T is the Hecke algebra.
#
V = MS.ambient_module()
e = V([0,oo])  # the winding element
e

-(1,0)

Phi(e)

(-1/7, 0, 0, 0)

V.sturm_bound()
PhiTe = span([Phi(V.hecke_operator(n)(e)) for n in [1..8]], ZZ)
PhiTe

8 Free module of degree 4 and rank 2 over Integer Ring Echelon basis matrix: [1/7 0 0 0] [ 0 2 -2 2]

PhiTe.index_in(PhiTe.saturation())

2/7

So for this 2-dimensional abelian variety of "level 43", $$\frac{L(A_f,1)}{\Omega_{A_f}} = \frac{2}{7}.$$

T = A.rational_torsion_subgroup()
T.cardinality()  # open problem

Error in lines 2-2 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/categories/finite_enumerated_sets.py", line 149, in _cardinality_from_iterator for _ in self: File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/structure/gens_py.py", line 50, in abelian_iterator G = M.gens() File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/finite_subgroup.py", line 575, in gens B = [self.element_class(self, v) for v in self.lattice().basis() if v.denominator() > 1] File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/torsion_subgroup.py", line 240, in lattice raise NotImplementedError("unable to compute the rational torsion subgroup in this case (there is no known general algorithm yet)") NotImplementedError: unable to compute the rational torsion subgroup in this case (there is no known general algorithm yet)

# but...
T.divisor_of_order()

7

T.multiple_of_order()

14

Booh regarding the below. Implementing this very algorithm is what got me into Magma in the late 1990s... http://wstein.org/papers/compgrp/

I spent a huge amount of time (starting at an Arizona Winter School with Robert Bradshaw in maybe 2008) implementing the hard part of this algorithm, which involves subtle computations with quaternion algebra "class sets"... but evidently never got to put things together!

Just seeing this half-finished situation makes me so annoyed.

A.tamagawa_number(43)

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/structure/parent.pyx", line 859, in sage.structure.parent.Parent.__getattr__ (/projects/sage/sage-6.10/src/build/cythonized/sage/structure/parent.c:8131) attr = getattr_from_other_class(self, self._category.parent_class, name) File "sage/structure/misc.pyx", line 253, in sage.structure.misc.getattr_from_other_class (/projects/sage/sage-6.10/src/build/cythonized/sage/structure/misc.c:1667) raise dummy_attribute_error AttributeError: 'ModularAbelianVariety_newform_with_category' object has no attribute 'tamagawa_number'

However, in this case, there is a theorem of Mazur, (from Modular curves and the Eisenstein Ideal), that the Tamagawa number at $p=43$ is the numerator of $(p-1)/12$, i.e.,

numerator((43-1)/12)

7

So BSD is that Sha has order between blah and blah. Work it out on the blackboard.

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