MSRI Modular Forms Summer Workshop Problem Book

Organizer: William Stein

July 31-Aug 12, 2006

Abstract

This MSRI workshop will be focused on making progress toward the problems listed in this document. The goal of the workshop will be to produce papers, software, and web pages related to each of the problems. These documents should be of interest to active researchers in modular forms (target audience: the workshop speakers, and, e.g., Barry Mazur, Karl Rubin, Ralph Greenberg, Matt Emerton, Frank Calegari, graduate students, etc.)

Instead of dividing the participants into groups with leaders the five invited lecturers, we will instead be dividing into groups who will attack 8 groups of research problems. These problems are not meant to span the whole gambit of problems about computing with modular forms – they are a necessarily biased selection of problems.

Everybody should find their email address listed in the group for at least two problems below. (If not, email wstein@gmail.com.) You should focus primarily on the problems you’ve been assigned to. You’ll probably discover that you make progress on one problem, but not the other, which is perfectly fine – that’s why you have two problems. Also, everybody should feel welcome to look at or help with any problem.

Do not be unduly discouraged if you find every problem difficult – every single one is an unsolved problem or project, as far as we know.

Most problems below that say “implement in SAGE” can be interpreted to mean “implement using only free open source software,” since it is very easy for SAGE to include any such programs. Moreover, by “algorithm” below we usually mean a “practical algorithm”, i.e., something that could be implemented and actually used to obtain interesting insight.

Contents

1 Characteristic Polynomials of Hecke Operators 3

1.1 Level 1 Weight 24 . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Preliminary theorems . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Main Result and its Proof . . . . . . . . . . . . . . . . . . 4

1.2 Higher Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 High Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Modular Forms Modulo p 7 2.1 Big images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Non-liftable weight one modular forms over 𝔽_{p} . . . . . . . . . . 8

3 Hilbert Modular Forms 9 3.1 Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Investigate the Birch and Swinnteron-Dyer conjecture . . . . . . 9

4 Computing with Classical Modular Forms 10 4.1 Compute Every Elliptic Curve of Conductor 234446 . . . . . . . 10 4.2 Find all elliptic curves of conductor ≤ 234446 . . . . . . . . . . . 11 4.3 Weight 144169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.4 A Problem About Bernoulli Numbers . . . . . . . . . . . . . . . 12 4.5 Half Integral Weight Modular Forms . . . . . . . . . . . . . . . . 13

5 Non-classical Modular Forms 14 5.1 Hilbert-Siegel Modular Forms . . . . . . . . . . . . . . . . . . . . 14

6 p-adic Heights 16 6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2 An Example: Computing E_{2}((37a, 5)) . . . . . . . . . . . . . . . . 19

6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.2.2 Example: Computing the Matrix of Frobenius for 37A at

p = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.2.3 Source Code . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2.4 Future Problems . . . . . . . . . . . . . . . . . . . . . . . 26 7 Level Raising and Lowering Modulo p^{n} 28

7.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8 Invariants of Modular Abelian Varieties 30 8.1 Endomorphism ring of Ā/𝔽_{p} . . . . . . . . . . . . . . . . . . . . . 30 8.2 Endomorphism Rings over Number Fields . . . . . . . . . . . . . 32 8.3 Component Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.4 Mordell-Weil Groups and Torsion subgroups . . . . . . . . . . . . 33 8.5 Analytic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.6 The Manin constant . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.7 Birch and Swinnerton-Dyer Conjectures . . . . . . . . . . . . . . 35

8.7.1 Refined Conjectures . . . . . . . . . . . . . . . . . . . . . 35 8.7.2 Intrinsic Characterization of Optimality . . . . . . . . . . 35

8.8 The Shafarevich-Tate Group . . . . . . . . . . . . . . . . . . . . . 36 8.8.1 Verifying the Full Conjecture for Elliptic Curves . . . . . 36 8.8.2 Finiteness of � . . . . . . . . . . . . . . . . . . . . . . . 36

Chapter 1

Characteristic Polynomials of Hecke Operators

Group Members:

aftab@brandeis.edu
bviray@math.berkeley.edu
dietelb@math.oregonstate.edu
gabor.wiese@mathematik.uni-regensburg.de
hiar0002@umn.edu
jallotta@math.northwestern.edu
jedwab@usc.edu
jhower@math.uga.edu
justin@mac.com
koot@u.washington.edu
kiwisquash@gmail.com

1.1 Level 1 Weight 24

Gather data and theoretical results about the following question, which Ralph Greenberg asked (though surely other people have asked it):

Problem 1.1.1. Is the characteristic polynomial of every Hecke operator T_{p} irreducible on the 2-dimensional space of cusp forms of level 1 and weight 24? (Stein checked that it is for all primes up to 800 and for p=144169.)

The rest of this section was written by Koopa Koo, and explains some results related to (1.1.1).

The goal of this section is to explain how to apply Chebotarev’s density theorem to obtain density result about irreducibility of the characteristic polynomial of the Hecke operators attached to the weight 24 cusp forms of level 1, S_{24}(Γ_{0}(1)).

I would like to thank my advisor Prof. Ralph Greenberg for suggesting the problem with helpful suggestions, Prof. Gabor Wiese for his helpful suggestions, and my greatest thanks to Prof. William Stein, who advised the project and provided many inspiring ideas.

1.1.1 Preliminary theorems

In this section, I will state the theorems I will use to prove the result mentioned in the abstract.

Theorem 1.1.2 (Chebotarev Density Theorem). Let L be a finite Galois extension of the number field K, with Gal(L/F) = G. Let X ⊂ G, stable under conjugation. Let P_{X} be the set of places v ∈ Σ_{K}, unramified in L, such that the Frobenius class Frob_{v} is contained in X. Then P_{X} has density equal to |X|/|G|.

Corollary 1.1.3. Suppose L/K is a Galois extension of number fields with G = Gal(L/K). Then the set of primes p ∈ K which splits completely has density 1/[L : K].

Theorem 1.1.4 (Ribet’s Big Image Theorem). Suppose f is a non-CM weight k level 1 eigenform form. Then the image of the mod ℓ representation is as big as possible for all but finitely many ℓ.

Theorem 1.1.5 (Deligne). For each newform and for every prime ℓ, there exists a continuous Galois representation:

ρ_{ℓ} : Gal(K_{ℓ}/ℚ) → GL_{2}(ℤ_{ℓ}),

(where K_{ℓ} is the maximal extension of Q unramified outside of ℓ) such that the characteristic polynomial of ρ_{ℓ}(Frob_{p}), p ≠ ℓ equals t^{2} − a(p)t + p^{k−1}, where a(p) is the p-th coefficient of f, and k its weight.

1.1.2 Main Result and its Proof

Theorem 1.1.6. Suppose f ∈ S_{24}(Γ_{0}(1)) is an eigenform. (i.e. f = q + a_{2}q^{2} + …) and a_{p} is a root of charpoly(T_{p}). Then the set

{p prime : charpoly(T_{p}) is reducible over ℚ}

has density 0.

Proof. Let f ∈ S_{24}(Γ_{0}(1)) be an eigenform, such that f = q +a^{2}q^{2} +…, and we know that a_{2} is a root of charpoly(T_{2}) and it is irreducible. Let K_{f} = ℚ(\sqrt{D}) be the quadratic field that is generated by the coefficients a_{n} of f.

Let

P_{ℓ} = {p prime : charpoly(T_{p}) is reducible over 𝔽_{ℓ}}.

Thus P_{ℓ} = \{p prime : \overline{a_{p}} ∈ 𝔽_{ℓ}\} since a_{p} is a root of charpoly(T_{p}), which in turn is equal to \{p prime : tr(\overline{ρ}(Frob_{p}))) ∈ 𝔽_{ℓ}\} by Deligne’s theorem, where \overline{ρ} : G_{ℚ} → GL_{2}(O_{K_{f}} /(ℓ)) is the mod ℓ Galois representation.

Note that \overline{a_{2}} ∈ 𝔽_{ℓ} iff charpoly(T_{2}) splits in 𝔽_{ℓ}[t] iff ℓ splits completely in O_{K_{f}}. By Chebotarev density theorem, the set of primes which split completely in O_{K_{f}} has density 1/2. This says the density of inert primes is 1/2.

Since f is a non-CM form, Ribet’s big image theorem says the projective mod ℓ Galois representation: proj(\overline{ρ}) : G_{ℚ} → PGL_{2}(O_{K_{f}} /(ℓ)) is surjective for all but finitely many primes ℓ. In particular, since there are infinitely many primes ℓ that are inert, we have

proj(\overline{ρ}) : G_{ℚ} → PGL_{2}(𝔽_{ℓ^{2}})

is surjective for infinitely many inert primes ℓ.

Now let G = Gal(\overline{ℚ}^{ker(proj(\overline{ρ}))}/ℚ), then |G| = | image(proj(\overline{ρ}))| = | PGL_{2}(𝔽_{ℓ^{2}})|, by Galois theory.

Let

F_{c} = \{p : [proj(\overline{ρ}(Frob_{p}))] = c\},

for each conjugacy class c ∈ PGL_{2}(𝔽_{ℓ^{2}}) whose trace is in 𝔽_{ℓ}. Here, [proj(\overline{ρ}(Frob_{p}))] denotes the conjugacy class of proj(\overline{ρ}(Frob_{p})).

Then by Chebotarev density theorem, we have the density of F_{c} equals |c|/|G|.

Now, P_{ℓ} = \{p : tr(\overline{ρ}(Frob_{p})) ∈ 𝔽_{ℓ}\} ⊂ ∪_{c}F_{c}, where c runs through all conjugacy classes of PGL_{2}(𝔽_{ℓ^{2}}) with trace in 𝔽_{ℓ}. The union is disjoint and we have:

density of P_{ℓ} ≤ ∑_{c} \frac{|c|}{|G|}.

The following lemma calculates the number of conjugacy classes c and the size of each c:

Lemma 1.1.7. There are three types of conjugacy classes c in PGL_{2}(𝔽_{ℓ^{2}}), which are:

I, (\begin{matrix}1 & 0 \\ 0 & a \end{matrix}) , (\begin{matrix}b & 1 \\ 0 & b \end{matrix}) ,

where a ≠ 1.

Furthermore, with the restriction of trace, and determinant being in the ground field 𝔽_{ℓ} there are 1, ℓ−2, ℓ−1 conjugagcy classes of the 1st, 2nd, and 3rd type respectively and the size of c of type 1, 2, 3 are 1, ℓ^{2} − 1, and (ℓ^{2} − 1)(ℓ^{2}) respectively.

Proof. With the restriction that trace and determinant are in the ground field, we have that 𝔽_{ℓ^{2}} contains all eigenvalues and we may apply Jordan Canonical form, then we have the classification into the three types at once. It is obvious that there are only 1 class that contains I. For the 2nd type, since determinant lies in the ground field implies a ∈ 𝔽_{ℓ}, we have ℓ − 2 choice because we are omitting 0 and 1. Finally, for the 3rd type, 2a ∈ 𝔽_{ℓ} implies a ∈ 𝔽_{ℓ} and that left ℓ − 1 choices for a by omitting 0. The size of c of 1st type is obviously 1 since everything commutes with I. By direct calculation, the set of matrices that commutes with a type 2 matrix and type 3 matrix looks like:

(\begin{matrix}1 & 0 \\ 0 & x \end{matrix}) , (\begin{matrix}x & y \\ 0 & z \end{matrix}) ,

respectively and there are ℓ^{2} − 1 and (ℓ^{2} − 1)(ℓ^{2}) of each respectively. This finishes the proof of the lemma.

By the lemma we have:

∑_{c} \frac{|c|}{|G|} = \frac{1 + (ℓ^{2} − 1)(ℓ − 2) + (ℓ − 1)(ℓ^{2} − 1)(ℓ^{2})}{(ℓ^{2} + 1)(ℓ^{4} − ℓ^{2})},

which goes to 0 as ℓ → ∞.

Hence density of P_{ℓ} = 0.

Finally, since

{p : charpoly(T_{p}) is reducible over ℚ} ⊂ {p : charpoly(T_{p}) is reducible over 𝔽_{ℓ}}.

We have the density of {p : charpoly(T_{p}) is reducible over ℚ} = 0. This concludes the proof.

Problem 1.1.8. I expect the same method generalizes to higher weight and higher level, by working with mod λ representations. Help me carry out this generalization and write up the details nicely.

1.2 Higher Level

The results in Section 1.1 strongly suggest that if f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(23)) is a newform, then asymptotically 0% of a_{p} are in ℚ (for p prime). If you actually look at the data, a large proportion of a_{p} do lie in ℚ for relatively small p; as p gets larger the proportion (appears to) tend to 0. In contrast for every newform f ∈ S_{2}(Γ_{0}(N)) of degree ≥ 3 that I (=Stein) investigated the proportion of a_{p} ∈ ℚ was very very small.

Problem 1.2.1. Explain the above observations about the proportion of a_{p} that lie in ℚ. Gather further numerical data.

1.3 High Weight

Farmer and James (see [CF99] and [FJ02]) verified that the characteristic polynomials of a very impressive range of Hecke operators T_{p,k} on level one modular forms are irreducible.

Problem 1.3.1. Greatly extend the range of computation carried out by Farmer and James. For example, verify that T_{2} acts irreducibly on S_{k}(1) for all k ≤ 4096.

Modular Forms Modulo p

Group Members:

citro@math.ucla.edu
clemesha@gmail.com
datta@math.umass.edu
gabor.wiese@mathematik.uni-regensburg.de
jallotta@math.northwestern.edu
jedwab@usc.edu
leyw@maths.usyd.edu.au
lxiao@mit.edu

These two projects certainly yield new results.

2.1 Big images

This section was written by Gabor Wiese.

To every mod p eigenform Deligne attaches a 2-dimensional odd ”mod p” Galois representation, i.e. a continuous group homomorphism

Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽}_{p}).

The trace of a Frobenius element at a prime l is for almost all l given by the l-th coefficient of the (normalised) eigenform. By continuity, the image of such a representation is a finite group.

Problem 2.1.1. Find group theoretic criteria that allow one (in some cases) to determine the image computationally.

Remark 2.1.2. (From Richard Taylor) Problem 2.1.1 seems to me straightforward. (Richard, Grigor, and Stein did something like this for elliptic curves over ℚ —

see http://modular.math.washington.edu/papers/bsdalg/.)

Problem 2.1.3. Implement in SAGE the algorithm of Problem 2.1.1.

Problem 2.1.4. Carry out systematic computations of mod p modular forms in order to find “big” images.

Like this one can certainly realise some groups as Galois groups over ℚ that were not known to occur before!

2.2 Non-liftable weight one modular forms over 𝔽_{p}

This section was written by Gabor Wiese.

This problem is closely connected to the ”big images” challenge, and could/should be treated in collaboration. Modular forms of weight 1 over 𝔽_{p} behave completely differently from forms of higher weights. One feature is that they are very often NOT reductions of holomorphic modular forms. In the course it will be explained how to compute modular forms of weight one. By looking at the image of a weight one form, one can often prove that it is such a non-liftable form. So far, there are many examples over 𝔽_{2}, but only one example for an odd prime, namely for p = 199.

Problem 2.2.1. Find examples in small odd characteristics!

Hilbert Modular Forms

Group Members:

aftab@brandeis.edu
cipra@math.ksu.edu
davidg@maths.usyd.edu.au
dembele@math.ucalgary.ca
jhb33@email.byu.edu
mateddy@ust.hk
muskat@math.colostate.edu
saha@caltech.edu
stephanie.jakus@gmail.com
ute@math.uni-sb.de
zhi@math.ucr.edu

3.1 Database

Problem 3.1.1. Create a free web-accessible database of Hilbert modular forms on real quadratic fields of small discriminants.

3.2 Investigate the Birch and Swinnteron-Dyer conjecture

Problem 3.2.1. Find equations for modular abelian surfaces of small levels and used them to investigate the Birch and Swinnerton-Dyer conjecture for real quadratic fields.

Remark 3.2.2. (From Richard Taylor) I would be very interested in this.

Computing with Classical Modular Forms

Group Members:

ccaranic@math.lsu.edu
dembele@math.ucalgary.ca
hiar0002@umn.edu
jenb@mit.edu
jhb33@email.byu.edu
jhower@math.uga.edu
jonhanke@math.duke.edu
kohel@maths.usyd.edu.au
mayab@math.lsu.edu
muskat@math.colostate.edu
robertwb@math.washington.edu
stephanie.jakus@gmail.com
wstein@gmail.com

4.1 Compute Every Elliptic Curve of Conductor 234446

Problem 4.1.1. Find with computational proof every elliptic curve over ℚ of conductor 234446.

Possible strategies:

1. Run Cremona’s program (i.e., use modular symbols and linear algebra). This program is freely available, but I don’t think anybody but Cremona actually uses it, so... This may or may not work, depending on constraints in the program. Use sage.math.washington.edu which has 64GB RAM.

2. Use the Mestre method of graphs applied to level 2 · 117223 to compute a sparse matrix for T_{2}. This will result in a very sparse matrix. Find the dimensions of the eigenspaces of T_{2} with eigenvalues −2, −1, 0, 1, 2, possibly using Wiedemann’s algorithm, or sparse linear algebra (or ???). Do the same for T_{3}, T_{5}, etc., if necessary.

3. Create an algorithm based on Dembelle-Stein-Kohel’s ideas, i.e., compute in the quaternion algebra ramified at 2 and ∞ with auxiliary level 117223. This will lead to the same linear algebra problem as we get with Mestre’s method.

I’m certain 2 or 3 above will succeed, since Andrei Jorza, Jen Balakrishnan, and I did something similar (for prime level) 2 years ago successfully.

Remark 4.1.2. (From Mark Watkins:)

Problem 4.1.1 was done by Cremona when I visited him last November. There are 3 curves of this conductor, two of rank 3, and one of rank 4. All are in the SW ECDB.

234446 [2,117223] 4 8.943847 1 +334976
[1,-1,0,-79,289] [2,1] X 1
234446 [2,117223] 3 9.848943 1 +82752
[1,1,0,-696,6784] [6,1] X 1
234446 [2,117223] 3 19.244917 1 +229824
[1,1,1,-949,-7845] [18,1] X 1

Remark 4.1.3. (From John Cremona:)

I ran level 234446 some time ago! It is not true that the curve of rank 4 is the only one! There are two others and they both have rank 3!

234446  a       1       [1,1,0,-696,6784]       3       1       0
234446  b       1       [1,-1,0,-79,289]        4       1       0
234446  c       1       [1,1,1,-949,-7845]      3       1       0

I emailed you about this on 20 November 2005....

–so your Problem 4.1.1 needs to be changed....but 4.2 still remains, of course! John

4.2 Find all elliptic curves of conductor ≤ 234446

Andrei Jorza, Jen Balakrishnan, and I verified that the Stein-Watkins tables

   http://modular.math.washington.edu/Tables/ecdb/

are complete for prime conductors p < 234446. This proved that the smallest conductor of a rank 4 elliptic curve is not prime. Is the smallest conductor 234446? To find out, one has to compute every elliptic curve (up to isogeny) of conductor N ≤ 234446. Cremona has computed every curve of conductor ≤ 130000, and much more about each curve (e.g., pretty much everything we know how to compute about a curve).

Problem 4.2.1. Determine all elliptic curves over ℚ of conductor ≤ 234446. By “determine” this could be man finding just the first few a_{p} = p+1−#E(𝔽_{p}) for each curve, not the actual equation.

The Stein-Watkins tables

   http://modular.math.washington.edu/Tables/ecdb/

contains a “substantial chunk” of the curves of conductor ≤ 234446. Challenge 4.2.1 amounts to finding the number (and some info about) the curves that are missing from Stein-Watkins in the range of conductors

130000 < N ≤ 234446.

4.3 Weight 144169

Barry Mazur’s 144169 problem (see the two-page pdf that Mazur sent me).

Remark 4.3.1. (From Barry Mazur.)

I’ve just glanced (fast) through the problem book and I wanted to send you a note telling you that I think it is wonderful. It really seems enticing, rich, friendly, and vastly interesting. About the 144169 problem, there was a bit of discussion about it and related things last Spring (Kevin Buzzard, and Robert Pollack, in particular, had ideas). The more general question behind the 144169 example is to consider 𝕋 = 𝕋_{p} the p-adic (Hida) Hecke algebra (say, of tame level 1, for starters) as a Λ-algebra and to form D = discriminant of the finite flat Λ-algebra 𝕋, so that D is in Λ (i.e., is an Iwasawa function). We want to know something about the basic invariants of D, e.g., its ”λ-invariant ” in each of the p−1 discs that form the rigid-analytic space underlying Λ = ℤ_{p}[[ℤ_{p}^{∗}]] and more specifically, we want to know something about the placement of the zeroes of D, if there are any. With p=144169 and the 24th disc, since that part of 𝕋 is quadratic over that part of Λ, there could be some zeroes, so the question is: are there some, and how many? If I remember right, Kevin had an idea about how to quickly compute this and Pollack had an idea of how—in the context of some tame level—to get examples for low primes like p = 3 and p = 5, where D has some zeroes.

4.4 A Problem About Bernoulli Numbers

This section was written by Ralph Greenberg.

Definition 4.4.1 (Irregular Prime). A prime p is said to be irregular if p divides the numerator of a Bernoulli number B_{j}, where 2 ≤ j < p − 1 and j is even. (For odd j, one has B_{j} = 0.)

The index of irregularity for a prime p is the number of such j.’s There is considerable numerical data concerning the statistics of irregular primes - the proportion of p < x which are irregular or which have a certain index of irregularity. (See Irregular primes and cyclotomic invariants to four million, Buhler et al., in Math. of Comp., vol. 61, (1993), 151-153.)

Let

C_{j} = \frac{1}{p} (\frac{B_{j}}{j} − \frac{B_{j+p−1}}{j + p − 1})

for each j as above. According to the Kummer congruences, C_{j} is a p-integer, i.e., its denominator is not divisible by p. But its numerator could be divisible by p. This happens for p = 13 and j = 4.

Problem 4.4.2. Obtain numerical data for the divisibility of the numerator of C_{j} by a prime p analogous to that for the B_{j}’s.

Motivation: It would be interesting to find an example of a prime p and an index j (with 2 ≤ j < p − 1, j even) such that p divides the numerator of both B_{j} and C_{j}. Then the p-adic L-function for a certain even character of conductor p (namely, the p-adic valued character ω^{j}, where ω is the character characterized by ω(n) ≡ n (mod pℤ_{p}) for n ∈ ℤ) would have at least two zeros. No such example exists for p < 16, 000, 000. The p-adic L-functions for those primes have at most one zero. If the statistics for the C_{j}’s are similar to those for the B_{j}’s, then a probabilistic argument would suggest that examples should exist.

Problem 4.4.3. Computation of B_{j} for a specific j is very efficient in PARI, hence in SAGE via the command bernoulli. Methods for computation of B_{j} (mod n) for a large range of j are described in Irregular primes and cyclotomic invariants to four million, Buhler et al. Implement the method of Buhler et al. in SAGE.

4.5 Half Integral Weight Modular Forms

Problem 4.5.1. Implement the algorithm in Basmaji’s Ph.D. thesis and make tables that are easy to use in SAGE. (I think I have an implementation somewhere in MAGMA of the algorithm – Stein.)

Non-classical Modular Forms

Group Members:

kiwisquash@gmail.com
ute@math.uni-sb.de
saha@caltech.edu
davidg@maths.usyd.edu.au
mayab@math.lsu.edu
naeem@math.ksu.edu
zhi@math.ucr.edu
jared@math.berkeley.edu
gunnells@math.umass.edu
nathan@math.ucla.edu

5.1 Hilbert-Siegel Modular Forms

Problem 5.1.1. Use quaterion algebras and Brandt modules to find examples of Hilbert-Siegel modular forms.

If possible, also study the Galois representations corresponding to Siegel modular forms.

Skoruppa has done work on tables of Siegel modular forms:

   http://wotan.algebra.math.uni-siegen.de/~modi/

Problem 5.1.2. Make the data available in Skoruppa’s tables easily accessible in SAGE. If there are dimension formulas, implement them in SAGE. If there are tables, download them all and put them into SAGE. Be sure to appropriately acknowledge Skorrupa’s contribution.

Remark 5.1.3. (From Richard Taylor) I would be very interested in this.

You can also compute on unitray groups (Lassine has been doing something along these lines and Kevin was planning to too.)

If you are going to work with siegel modular forms of level > 1 (which you should do!), you have to be very careful what levels you use. I have never understood this properly, but I think it does not suffice to work with what people call Γ_{0}(N) - defined similarly to Γ_{0}(N) for GL_{2} but using 2 × 2 blocks.

Remark 5.1.4. (From John Cremona:)

Why is there not a Chapter similar to this but about imaginary quadratic fields? [By the way, I once mentioned that I would have a new PhD student starting in September who would implement higher weight modular symbols over such fields. But he has decided to stay in Cambridge, with Tom Fisher, so that project remains open.]

p-adic Heights

Group Members:

burhanud@usc.edu
ccaranic@math.lsu.edu
citro@math.ucla.edu
darnall@math.wisc.edu
dietelb@math.oregonstate.edu
dmharvey@fas.harvard.edu
jenb@mit.edu
kohel@maths.usyd.edu.au
koot@u.washington.edu
limburgs@math.oregonstate.edu
lxiao@mit.edu
robertwb@math.washington.edu
wstein@gmail.com

6.1 Background

The following is adapted from the introduction to [MST06].

Let p be an odd prime number, and E an elliptic curve over a global field K that has good ordinary reduction at p. Let L be any (infinite degree) Galois extension with a continuous injective homomorphism ρ of its Galois group to ℚ_{p}. To the data (E, K, ρ), one associates^{1} a canonical (bilinear, symmetric) (p-adic) height pairing

( , )_{ρ} : E(K) × E(K) ⟶ ℚ_{p}.

Such pairings are of great interest for the arithmetic of E over K, and they arise specifically in p-adic analogues of the Birch and Swinnerton-Dyer conjecture.^{2}

1See [MT83], [Sch82] [Sch85], [Zar90], [Col91], [Nek93], [Pla94], [IW03], and [Bes04].

2See [Sch82], [Sch85] [MT83], [MT87], [PR03]. See also important recent work of Jan Nekovář.

The goal of this project is to investigate some computational questions regarding p-adic height pairings. The main stumbling block to computing them efficiently is in calculating, for each of the completions K_{v} at the places v of K dividing p, the value of the p-adic modular form E_{2} associated to the elliptic curve with a chosen Weierstrass form of good reduction over K_{v}.

The paper [MST06] contains an algorithm for computing these quantities (for K = ℚ), i.e., for computing the value of E_{2} of an elliptic curve (that builds on the works of Katz and Kedlaya listed in our bibliography).

Problem 6.1.1. Implement the full algorithm in SAGE. (It is completely implemented in MAGMA already.)

The paper [MST06] also discusses the p-adic convergence rate of canonical expansions of the p-adic modular form E_{2} on the Hasse domain, where for p ≥ 5 we view E_{2} as an infinite sum of classical modular forms divided by powers of the (classical) modular form E_{p−1}, while for p ≤ 5 we view it as a sum of classical modular forms divided by powers of E_{4}.

Problem 6.1.2. Compute p-adic heights for elliptic curves in families, e.g., for curves over ℚ(t). Interpret the result in terms of log convergence.

We were led to our fast method of computing E_{2} by our realization that the more naive methods, of computing it by integrality or by approximations to it as function on the Hasse domain, were not practical, because the convergence is “logarithmic” in the sense that the nth convergent gives only an accuracy of log_{p}(n).

The reason why this constant E_{2} enters the calculation is because it is needed for the computation of the p-adic sigma function [MT91], which in turn is the critical element in the formulas for height pairings.

For example, let us consider the cyclotomic p-adic height pairing in the special case where K = ℚ and p ≥ 5.

If G_{ℚ} is the Galois group of an algebraic closure of ℚ over ℚ, we have the natural surjective continuous homomorphism χ : G_{ℚ} → ℤ_{p}^{∗} pinned down by the standard formula g(ζ) = ζ^{χ(g)} where g ∈ G_{ℚ} and ζ is any p-power root of unity. The p-adic logarithm log_{p} : ℚ_{p}^{∗} → (ℚ_{p}, +) is the unique group homomorphism with log_{p}(p) = 0 that extends the homomorphism log_{p} : 1 + pℤ_{p} → ℚ_{p} defined by the usual power series of log(x) about 1. Explicitly, if x ∈ ℚ_{p}^{∗}, then

log_{p}(x) = \frac{1}{p − 1} · log_{p}(u^{p−1}),

where u = p^{− ord_{p}(x)}·x is the unit part of x, and the usual series for log converges at u^{p−1}.

The composition (\frac{1}{p} · log_{p}) ∘ χ is a cyclotomic linear functional G_{ℚ} → ℚ_{p} which, in the body of our text, will be dealt with (thanks to class field theory) as the idele class functional that we denote ρ_{ℚ}^{cycl}.

Let E denote the Néron model of E over ℤ. Let P ∈ E(ℚ) be a non-torsion point that reduces to 0 ∈ E(𝔽_{p}) and to the connected component of E_{𝔽_{ℓ}} at all primes ℓ of bad reduction for E. Because ℤ is a unique factorization domain, any nonzero point P = (x(P), y(P)) ∈ E(ℚ) can be written uniquely in the form (a/d^{2}, b/d^{3}), where a, b, d ∈ ℤ, gcd(a, d) = gcd(b, d) = 1, and d > 0. The function d(P) assigns to P this square root d of the denominator of x(P).

Here is the formula for the cyclotomic p-adic height of P, i.e., the value of

h_{p}(P) := −\frac{1}{2}(P, P)_{p} ∈ ℚ_{p}

where ( , )_{p} is the height pairing attached to G_{ℚ} → ℚ_{p}, the cyclotomic linear functional described above:

h_{p}(P) = \frac{1}{p} · log_{p} (\frac{σ(P)}{d(P)}) ∈ ℚ_{p}.  (6.1.1)

Here σ = σ_{p} is the p-adic sigma function of [MT91] associated to the pair (E, ω). The σ-function depends only on (E, ω) and not on a choice of Weierstrass equation, and behaves like a modular form of weight −1, that is σ_{E,cω} = c·σ_{E,ω}. It is “quadratic” the sense that for any m ∈ ℤ and point Q in the formal group E^{f}(\overline{ℤ}_{p}), we have

σ(mQ) = σ(Q)^{m^{2}} · f_{m}(Q),  (6.1.2)

where f_{m} is the mth division polynomial of E relative to ω (as in [MT91, App. 1]). The σ-function is “bilinear” in that for any P, Q ∈ E^{f}(ℤ_{p}), we have

\frac{σ(P − Q) · σ(P + Q)}{σ^{2}(P) · σ^{2}(Q)} = x(Q) − x(P).  (6.1.3)

See [MT91, Thm. 3.1] for proofs of the above properties of σ.

The height function h_{p} of (6.1.1) extends uniquely to a function on the full Mordell-Weil group E(ℚ) that satisfies h_{p}(nQ) = n^{2}h_{p}(Q) for all integers n and Q ∈ E(ℚ). For P, Q ∈ E(ℚ), setting

(P, Q)_{p} = h_{p}(P) + h_{p}(Q) − h_{p}(P + Q),

we obtain a pairing on E(ℚ). The p-adic regulator of E is the discriminant of the induced pairing on E(ℚ)_{/ tor} (well defined up to sign), and we have the following standard conjecture about this height pairing.

Conjecture 6.1.3. The cyclotomic height pairing ( , )_{p} is nondegenerate; equivalently, the p-adic regulator is nonzero.

Problem 6.1.4. Gather substantial experimental evidence for Conjecture 6.1.3.

Remark 6.1.5. Height pairings attached to other p-adic linear functionals can be degenerate; in fact, given an elliptic curve defined over ℚ with good ordinary reduction at p, and K a quadratic imaginary field over which the Mordell-Weil group E(K) is of odd rank, the p-adic anticyclotomic height pairing for E over K is always degenerate.

Problem 6.1.6. Implement in SAGE the algorithm from Challenge 6.2.7 (see below).

Problem 6.1.7. Use Challenge 6.1.6 to do a massive computation for Mazur and Rubin related to their longterm explorations of universal norms.

The p-adic σ function is the most mysterious quantity in (6.1.1). There are many ways to define σ, e.g., [MT91] contains 11 different characterizations of σ! We now describe a characterization that leads directly to a (slow!) algorithm to compute σ(t). Let

x(t) = \frac{1}{t^{2}} + ⋯ ∈ ℤ_{p}((t))  (6.1.4)

be the formal power series that expresses x in terms of the local parameter t = −x/y at infinity. The following theorem, which is proved in [MT91], uniquely determines σ and c.

Theorem 6.1.8. There is exactly one odd function σ(t) = t + ⋯ ∈ tℤ_{p}[[t]] and constant c ∈ ℤ_{p} that together satisfy the differential equation

x(t) + c = −\frac{d}{ω} (\frac{1}{σ} \frac{dσ}{ω} ) ,  (6.1.5)

where ω is the invariant differential dx/(2y + a_{1}x + a_{3}) associated with our chosen Weierstrass equation for E.

Remark 6.1.9. The condition that σ is odd and that the coefficient of t is 1 are essential.

In (6.1.1), by σ(P) we mean σ(−x/y), where P = (x, y). We have thus given a complete definition of h_{p}(Q) for any point Q ∈ E(ℚ) and a prime p ≥ 5 of good ordinary reduction for E.

6.2 An Example: Computing E_{2}((37a, 5))

This section was written by Jennifer Balakrishnan.

We consider the algorithm of Kedlaya [Ked01] in the case of elliptic curves, as illustrated with an explicit example. In so doing, we compute the matrix of absolute Frobenius for the elliptic curve 37A at p = 5, a quantity essential to the computation of its 5-adic cyclotomic height pairing [MST06].

6.2.1 Introduction

Let E be the elliptic curve y^{2} = Q(x) over ℤ_{p}, given in Weierstrass form, and suppose p ≥ 5 is a prime of good ordinary^{3} reduction. Kedlaya’s algorithm^{4} [Ked01], [Ked04] employs Monsky-Washnitzer cohomology of the affine curve E \ O to compute the zeta function of its reduction over finite fields. While the computation of zeta functions for elliptic curves is best done with algorithms that take into account the group structure, we note that Kedlaya’s algorithm was formulated in the more general context of hyperelliptic curves and is of great interest for curves of genus greater than 1. However, more pertinent to our situation is the fact that Kedlaya’s algorithm allows us to compute the matrix of absolute Frobenius on Monsky-Washnitzer cohomology, and it is this matrix that allows for fast high-precision computation of p-adic heights using the recent algorithm of Mazur, Stein, and Tate [MST06].

3Kedlaya’s algorithm only requires good reduction; however, the application to computation of p-adic heights requires that p be ordinary as well.

4The reader is also encouraged to see the exposition of Edixhoven [Edi], which heavily influenced the writing of these notes.

We focus on the computation of this matrix. Details omitted here can be found in the aforementioned papers.

Let C\overline{Q} denote the affine curve over 𝔽_{p} cut out by the equation y^{2} = \overline{Q}(x). Consider C′_{Q} = C_{Q} \ {zeros of y}, and let

A = ℚ_{p}[x, y, z]/(y^{2} − Q(x), yz − 1)

denote the coordinate ring of C′_{Q} over ℚ_{p}. Recall that the hyperelliptic involution

ι : (a, b) ↦ (a, −b)

gives us an automorphism of the curves C_{Q} and C′_{Q}. This, in turn, induces automorphisms ι^{∗} of algebraic de Rham cohomology H^{1}(C′_{Q}) and H^{1}(C_{Q}), decomposing them into eigenspaces on which ι^{∗} acts as the identity and −1, respectively. In particular,

H^{1}(C′_{Q}) = H^{1}(C′_{Q})^{+} ⊕ H^{1}(C′_{Q})^{−}.

The ℚ_{p}-vector space H^{1}(C′_{Q})^{−} is spanned by the classes of differentials

{[zdx], [xzdx]} .

However, the underlying coordinate ring A does not admit the proper lift of Frobenius. To remedy this, we replace A by the dagger ring

A^{†} = \{∑_{i,j} a_{i,j}x^{i}y^{j} : a_{i,j} ∈ ℚ_{p}, lim inf_{|j|→∞} \frac{v_{p}(a_{i,j})}{|j|} > 0\} .

The de Rham complex of A^{†} is given by

d : A^{†} ⟶ A^{†} \frac{zdx}{2},
∑_{i,j} a_{i,j}x^{i}z^{j} ↦ ∑_{i,j} a_{i,j}d(x^{i}z^{j})

= ∑_{i,j} a_{i,j}(2ix^{i−1}z^{j−1} − jx^{i}Q′z^{j+1})\frac{zdx}{2}.

We denote the cohomology groups of this complex by H_{MW}^{i}(C′_{Q}), and as before, they are ℚ_{p}-vector spaces split into eigenspaces by the hyperelliptic involution. Perhaps more important is that passing from A to A^{†} does not change the presentation of cohomology, and thus we work with H_{MW}^{1}(C′_{Q})^{−} and its basis zdx and xzdx.

We compute the action of Frobenius on H_{MW}^{1}(C′_{Q})^{−} by computing its action on the basis elements. Begin by letting

G(x) = \frac{Frob_{p}(Q(x)) − (Q(x))^{p}}{p} .

We have that

F_{p,i} := Frob_{p} (x^{i}zdx) = ∑_{0≤k<M} (\binom{−1/2}{k}p^{k+1}G^{k}x^{p(i+1)−1}z^{(2k+1)p−1}) zdx,

as an element of ℤ_{p}[x, y, z]/(y^{2} −Q(x), yz −1), with a precision of N digits. The constant N determines the number of digits of precision of the p-adic height to be computed (i.e., modulo p^{N}), and M is the smallest integer such that

M − ⌊log_{p}(2M + 1)⌋ ≥ N.

As the two differentials zdx and xzdx span H_{MW}^{1}(C′_{Q})^{−}, we must now be able to write an arbitrary element in (A^{−})^{†} \frac{zdx}{2} (where A^{−} = ⨁_{0≤i<3,j≡1(2)} ℚ_{p}x^{i}y^{j}) as a linear combination of d(x^{i}z^{j}), zdx, and xzdx. With this in mind, we employ a reduction algorithm. For the purposes of this reduction, the following definition is helpful:

Definition 6.2.1. Given a multivariate polynomial f(x, y, z) in ℤ_{p}[x, y, z]/(y^{2}− Q(x), yz −1), the highest monomial of f is the one with smallest power of z and largest power of x.

Example 6.2.2. The highest monomial of

d(x^{i}z^{j}) = 2ix^{i−1}z^{j−1} − 3jx^{i+2}z^{j+1} − jx^{i}z^{j+1}

is x^{i−1}z^{j−1} if 1 ≤ i < 3 and x^{2}z^{j+1} if i = 0.

Here we outline the reduction algorithm. Begin by computing a list of differentials d(x^{i}z^{j}), where 0 ≤ i < 3 and j ≡ 1 (mod 2). Group the terms in Frob_{p}(x^{i}zdx) as (∑ c_{i,j}z^{j})zdx, where c_{i,j} ∈ ℤ_{p}[x] have degree less than or equal to 3.

If F_{p,i} has a term (x^{i}z^{j})zdx with j > 0, consider the term (c_{i,j}z^{j})zdx where j is maximal. Take the unique linear combination of the d(x^{k}z^{j−1}) such that when this linear combination is subtracted off of F_{p,i}, the resulting “F_{p,i}” no longer has terms of the form (x^{m}z^{j})zdx. Repeat this process until F_{p,i} (or, in more precise terms, the resulting “F_{p,i}” at each step minus linear combinations of differentials) has no terms (x^{m}z^{j})zdx with j > 0.

If F_{p,i} has terms with j ≤ 0, let (x^{m}z^{j})zdx be the term with the highest monomial of F_{p,i}. Let (x^{k}z^{l})zdx be the term such that d(x^{k}z^{l}) has highest term (x^{m}z^{j})zdx and subtract off the appropriate multiple of d(x^{k}z^{l}) such that the resulting F_{p,i} no longer has terms of the form (x^{m}z^{j})zdx with j ≠ 0. Repeat this process until the resulting F_{p,i} is of the form (a_{0i} + a_{1i}x) zdx.

Finally, we can read off the entries of the matrix F of absolute Frobenius:

F = ( \begin{matrix}a_{00} & a_{01} \\ a_{10} & a_{11} \end{matrix} ) .

Problem 6.2.3. Implement a highly optimized version of the reduction algorithm in SAGE for elliptic curves.

6.2.2 Example: Computing the Matrix of Frobenius for 37A at p = 5

Let p = 5 and consider the elliptic curve 37A, with minimal model y^{2} + y = x^{3} − x.

Step 1. Put the elliptic curve into Weierstrass form y^{2} = x^{3} + a_{4}x + a_{6}, via the transformation

a_{4} = −\frac{c_{4}}{2^{4} · 3},
a_{6} = −\frac{c_{6}}{2^{5} · 3^{3}} .

In our case, we obtain the curve

y^{2} = x^{3} − x + \frac{1}{4}.

Let

Q(x) = x^{3} − x + \frac{1}{4}.

Step 2. Fix the precision N and compute M. In our case, N = 2 and M = 3.

Step 3. Compute the action of Frobenius on the two differentials zdx and xzdx as an element of ℤ_{p}[x, y, z]/(y^{2} − Q(x), yz − 1), with a precision of N digits. Furthermore, group the terms of Frob_{p}(x^{i}zdx) as ∑(p^{k+1}c_{i,k,j}z^{j})zdx, where the c_{i,k,j} are in ℤ_{p}[x] of degree less than 3.

In our case, we compute

Frob_{5}(zdx) ≡ (5xz^{2} + (5x + 5x^{2})z^{4})zdx (mod 25)
Frob_{5}(xzdx) ≡ (10 + 10x + 5x^{3} + (20 + 5x + 15x^{2})z^{2} + (10 + 20x + 15x^{2})z^{4})zdx (mod 25).

Step 4. Now we must reduce the differentials. We want to write each of the

F_{5,i} = Frob_{5}(x^{i}zdx)

as

(a_{0i} + a_{1i}x) zdx + ∑ d(x^{i}y^{j}) = (a_{0i} + a_{1i}x) zdx

in H_{MW}^{1}(C′_{Q})^{−}. We begin with

F_{5,0} ≡ (5xz^{2} + (5x + 5x^{2})z^{4})zdx (mod 25)

and compute the appropriate list of differentials:

\begin{matrix}i & j & d(x^{i}z^{j}) (mod 25) \\ 0 & 1 & (13z^{2} + 11z^{2}x^{2})zdx \\ 1 & 1 & (12 + 16z^{2} + 24z^{2}x)zdx \\ 2 & 1 & (13x + 16z^{2}x + 24z^{2}x^{2})zdx \\ 0 & 3 & (14z^{4} + 8z^{4}x^{2})zdx \\ 1 & 3 & (9z^{2} + 23z^{4} + 22z^{4}x)zdx \\ 2 & 3 & (10z^{2}x + 23z^{4}x + 22z^{4}x^{2})zdx \end{matrix}

Thus we wish to write (5x+5x^{2})z^{4} as a linear combination of 14z^{4}+8z^{4}x^{2}, 23z^{4} +22z^{4}x, and 23z^{4}x+22z^{4}x^{2}, all modulo 25 (we may ignore the lower powers of z present in the differentials, as we will take care of them in the steps to come). We find that taking

F_{5,0} − 5d(z^{3}) − 10d(xz^{3}) − 20d(x^{2}z^{3}) (mod 25)

leaves us with

(10 + 5x)z^{2} zdx.

Now we wish to write (10+5x)z^{2} as a linear combination of 13z^{2} +11z^{2}x^{2}, 16z^{2} + 24z^{2}x, and 16z^{2}x + 24z^{2}x^{2}, modulo 25. We find that taking

(10 + 5x)z^{2} zdx − 10d(z) − 5d(xz) − 10d(x^{2}z)

leaves us with

(15 + 20x)zdx.

Next, we reduce

F_{5,1} ≡ (10+10x+5x^{3}+(20+5x+15x^{2})z^{2}+(10+20x+15x^{2})z^{4})zdx (mod 25).

Note that this has an x^{3} zdx term, so we take care of this first:

F_{5,1} − \frac{1}{3}d(x^{4}z) = (13+2x+(13+10x+7x^{2})z^{2} +(10+20x+15x^{2})z^{4}) zdx.

Now we proceed as in the case of F_{5,0}, and we wish to write (10 + 20x + 15x^{2})z^{4} as a linear combination of 14z^{4}+8z^{4}x^{2}, 23z^{4}+22z^{4}x, and 23z^{4}x+ 22z^{4}x^{2}, all modulo 25. We find that taking

(13+2x+13z^{2}+10z^{2}x+7z^{2}x^{2}+10z^{4}+20z^{4}x+15z^{4}x^{2})zdx−10d(z^{3})−15d(xz^{3})−5d(x^{2}z^{3}) leaves us with

(13 + 2x + (3 + 10x + 7x^{2})z^{2}) zdx.

Finally, we wish to write (3 + 10x + 7x^{2})z^{2} as a linear combination of 13z^{2} + 11z^{2}x^{2}, 16z^{2} + 24z^{2}x, and 16z^{2}x + 24z^{2}x^{2}, all modulo 25. We find that taking

(13 + 2x + (3 + 10x + 7x^{2})z^{2} zdx − 20d(z) − 23d(xz) − 13d(x^{2}z)

leaves us with

(12 + 8x)zdx.

Step 5. Now we form the matrix F of the reduced differentials, where each reduced differential gives us a column in the matrix of absolute Frobenius. In our

case, we have F = ( \begin{matrix}15 & 12 \\ 20 & 8 \end{matrix} ).

As a consistency check, we have that F has trace 23, which is a_{5} modulo 25 and determinant −120, which is p = 5 modulo 25.

6.2.3 Source Code

Below is the implementation in SAGE [SJ05]:

R.<y,z,x> = PolynomialRing(QQ,3)
S.<y,z,x> = R.quotient([x^3 -x+1/4 - y^2, y*z - 1])
Q = x^3 - x + S(1/4)
Q5 = (x^5)^3 - (x^5) + S(1/4)
Qprime = 3*x^2 - S(1)
G = S(1/5) * (Q5 - Q^5)
M = 3
p = 5
def F(i):
    a = 0
    for k in range(M):
        b = binomial(-1/2, k) * p^(k+1) * G^k * x^(p*(i+1)-1) * z^((2*k+1)*p-1)
        a += b
    return a
V = VectorSpace(QQ,30)
def coefficient_vector(f):
    v = V(0)
    for e, c in f.dict().iteritems():
        v[3*e[1] + e[2]] = c

return v
def d(i,j):
i = i
j = j
if i == 0:
   return S(1/2 * -j) * Qprime * z^(1+j)
return S(1/2) * (2*i*x^(i-1)*z^(j-1)-  j*x^i*Qprime*z^(j+1))
def Wspace():
dA = [coefficient_vector(reduce_mod(d(i, j),5)) \
                          for i in range(3) for j in range(1,5 +1) if j%2!=0]
H1 = [coefficient_vector(reduce_mod(S(1),5)), coefficient_vector(reduce_mod(x,5))]
B = V.subspace(dA).basis()
print len(dA), len(B)
print B
print H1
return V.subspace_with_basis(B + H1)
def reduce_mod(f, n):
"""
Return the reduction of f modulo n.
"""
R.<y,z,x> = PolynomialRing(IntegerModRing(n),3)
return R(sage_eval(str(f),  {'x':x, 'y':y, 'z':z}))
def rd(i,j,n):
return reduce_mod(d(i,j),n)
def coeff_list_of_pow(f, n, var_z_number=1, var_x_number=2, d=2):
"""
if f is a polynomial in n variables and z is the var_number-th
generator of the parent of f, this returns the degree-d list of
coefficients of z^n up to degree d.
"""
z = f.parent().gen(var_z_number)
c = f.coefficient(z^n)
x = f.parent().gen(var_x_number)
return [c.coefficient(x^i) for i in range(d+1)]
def lowest_d_matrix(j, n, p=5):
"""
Return the 3x3 matrix mod p of coefficients of the coefficients
of z^n in the d(i,j) for i=0,1,2.  Here j > 0.
"""
MS = MatrixSpace(GF(p), 3)

    v = []
    for i in range(3):
        v.append(coeff_list_of_pow(rd(i,j,p), n))
    return MS(v).transpose()
def find_lincomb_needed_to_subtract_off(f, n, p=5):
    j = n-1
    A = lowest_d_matrix(j,n,p)
    v = coeff_list_of_pow(f, n)
    w = Vector(GF(p), v)
    return A^(-1)*w
f0 = reduce_mod(F(0),25)
find_lincomb_needed_to_subtract_off(reduce_mod((1/5)*F(0),5), 4)
f01 = f0 - 5*rd(0,3,25)-10*rd(1,3,25)-20*rd(2,3,25)
find_lincomb_needed_to_subtract_off(reduce_mod((1/5)*(10*z^2+5*z^2*x),5),2)
f02 = f01 - 5*(2*rd(0,1,25) + 1*rd(1,1,25) + 2*rd(2,1,25))
f1 = reduce_mod(F(1),25)
f11 = f1-Mod(1/3,25)*reduce_mod(d(4,1),25)
find_lincomb_needed_to_subtract_off(reduce_mod((1/5)*(10*z^4+20*z^4*x+15*z^4*x^2),5),4)
f12 = f11 - 10*rd(0,3,25)-15*rd(1,3,25)-5*rd(2,3,25)
f13 = f12 - 20*rd(0,1,25)-23*rd(1,1,25)-13*rd(2,1,25)

6.2.4 Future Problems

There remain many exciting problems to consider in the computation of p-adic heights for elliptic curves. Below we outline a few:

Problem 6.2.4. Are there simplifications one could make to the above algorithm taking into account the fact that we’re working with elliptic curves (e.g., using group structure, etc.)? Should we expect that the matrix of Frobenius be easier to compute in the case of genus 1 curves?

Problem 6.2.5. A question of John Tate: how does the cyclotomic p-adic height pairing change for families of elliptic curves, e.g., y^{2} = x^{3} +tx+1? What about considering families with constant j-invariant? Non-constant j-invariant? Curves with complex multiplication? Curves without complex multiplication?

Remark 6.2.6. (From Christian Wuthrich) In my J. London Soc. article I computed some and proved that they are locally analytic. The height of a section can have zeroes as a rigid p-adic analytic function. If the conjecture on the non-degeneracy of the heights is true, then the zeroes are not in ℤ but in ℤ_{p}. But I have no interpretation of what they could mean.

Problem 6.2.7. Extend the above algorithm to implement the computation of p-adic anticyclotomic heights, using new ideas of Mazur in [MST06].

Problem 6.2.8. (From Christian Wuthrich.) For computational reasons it would be interesting to also include the primes 2 and 3. It should be possible to write a more complicated Kedlaya algorithm at least for 3. In shark I reprogrammed the original approximation algorithm (beware s_{2} is not exactly integral) for 2 and 3 so that I can compute 3-primary parts of Sha. Kato’s theorem is not known to me for p = 2.

Problem 6.2.9. (From Christian Wuthrich.) There is a well-defined supersingular theory explained by Perrin-Riou. The Kedlaya algorithm can be used to compute the p-adic heights also in this case. As I did it in shark. In shark there are also padic heights for multiplicative primes using Mazur-Tate-Teitelbaum. Additive primes are a bit harder, one needs to pass to an extension over which they become semi-stable. So this calls for p-adic height over arbitrary number fields.

Level Raising and Lowering Modulo p^{n}

Group Members:

cipra@math.ksu.edu
darnall@math.wisc.edu
datta@math.umass.edu
dmharvey@fas.harvard.edu
gabor.wiese@mathematik.uni-regensburg.de
holden@math.wisc.edu
jared@math.berkeley.edu
jonhanke@math.duke.edu
kane@math.wisc.edu
leyw@maths.usyd.edu.au
syazdani@math.berkeley.edu
wstein@gmail.com

Ribet’s theorems about level raising and level lowering have been central in a huge amount of modern work on modular forms. For example, they play a famous role in the proof of Fermat’s last theorem. You should read about these theorems somewhere. One introduction is [RS01].

Diamond (see [Dia95], etc.) and Diamond-Taylor (in their “Nonoptimal levels” paper), and Russ Mann in his Ph.D. thesis, have all also done important work related to level lowering and raising.

Unfortunately, it seems that nobody has proved or even formulated a conjectural analogue of these results for congruences modulo p^{n} between eigenforms. There is work about higher congruences in that comes up when studying p-adic modular forms (see, e.g., [Col03]).

Some^{1} have expressed doubt that there can even be a good level raising or lowering theorem modulo p^{n}.

1Dimitar Jetchev

Remark 7.0.10. (From Richard Taylor.) I share Jetchev’s pessimism. You can presumably come up with a criterion for a congruence to a new mod p^{n} eigenform (ie a homomorphism from the new part of the Hecke alg to ℤ/p^{n}ℤ). But as soon as n > 1 this does not imply that there is a characteristic 0 newform which reduces to this modulo p^{n}. Rather the mod p^{n} eigenform could result from several newforms congruent to the original form modulo p.

7.1 Problems

Let f be a newform in S_{2}(Γ_{0}(N)) (or, more generally, in S_{k}(Γ_{1}(N))). Let λ be a prime ideal in the ring generated by the Fourier coefficients of f. Let n ≥ 1 be a positive integer.

Problem 7.1.1. Let E be the elliptic curve 11a given by the equation

y^{2} + y = x^{3} − x^{2} − 10x − 20,

and let f = f_{E} = q − 2q^{2} − q^{3} + ⋯ ∈ S_{2}(Γ_{0}(11)) be the corresponding newform. For

r = 9, 27, 7^{2}

compute all newforms g ∈ S_{2}(Γ_{0}(11q)) with q < 500 prime and g ≡ f (mod r). Is there a pattern?

Problem 7.1.2. Formulate a level raising conjecture modulo λ^{n}. Provide computational and theoretical evidence.

Problem 7.1.3. Formulate a level lowering conjecture modulo λ^{n}. Provide computational and theoretical evidence.

Invariants of Modular Abelian Varieties

Group Members:

burhanud@usc.edu
bviray@math.berkeley.edu
holden@math.wisc.edu
justin@mac.com
kane@math.wisc.edu
kohel@maths.usyd.edu.au
limburgs@math.oregonstate.edu
mateddy@ust.hk
nathan@math.ucla.edu
syazdani@math.berkeley.edu
wstein@gmail.com

After understanding algorithms for computing modular forms, one can focus on arithmetic information associated to them, particularly to the weight 2 cusp forms for Γ_{0}(N). These correspond to isogeny classes of abelian varieties over ℚ which are factors of the Jacobian J_{0}(N). In weight 2 one can look at the invariants of a particular representative modular abelian variety, rather than the more abstract notion of a Galois representation.

The goal of this project is to develop algorithms and implementations to investigate as many as possible of the following invariants for general modular abelian varieties.

Half of this chapter was written by David Kohel.

8.1 Endomorphism ring of Ā/𝔽_{p}

Let A = A_{f} be a modular abelian variety over ℚ associated to a newform in S_{2}(Γ_{0}(N)). Let p be a prime of good reduction for A (so (N, p) = 1). Let Ā = A_{𝔽_{p}} be the reduction of the A modulo p, which is an abelian variety over 𝔽_{p}.

Problem 8.1.1. Compute the endomorphism ring End(Ā/𝔽_{p}).

The endomorphism ring of Ā/𝔽_{p} contains 𝕋[π_{p}] = ℤ[{α_{n}}][π_{p}], where α_{n} is the n-th coefficient of the cusp form f of A, and the Frobenius endomorphism π_{p} satisfies π_{p}^{2} − α_{p}π_{p} + p = 0. If Ā is ordinary (i.e. has p-rank g = dim(A)), then

𝕋[π_{p}] ⊆ End(Ā) ⊆ O_{K}

where K = 𝕋[π_{p}] ⊗ ℚ and O_{K} is its maximal order. These reductions modulo p are CM abelian varieties, but in general only the real subring 𝕋 generated by the trace terms lift back to the modular abelian variety over ℚ.

Note that the invariant End(Ā) is an invariant of the isomorphism class, but not the isogeny class, of A. For instance the isogeny class of elliptic curves of conductor 57 denoted 57C by Cremona, consists of two curves:

\begin{matrix}E_{1} : y^{2} + y = x^{3} + x^{2} + 20x − 32, \\ E_{2} : y^{2} + y = x^{3} + x^{2} − 4390x − 113432, \end{matrix}

such that there exists a 5-isogeny ϕ : E_{1} → E_{2} between them. This induces isogenies on the reductions ϕ : Ē_{1} → Ē_{2}, from which one concludes, for each p, that either 5 is a split or ramified prime in O_{K}, or that 5 divides the index [O_{K} : ℤ[π_{p}]], and the two local endomorphism rings differ by index 5:

\frac{[O_{K} : End(Ē_{1})]}{[O_{K} : End(Ē_{2})]} ∈ \{5^{−1}, 5\}.

If we consider among the first 1000 primes those for which (5) is inert in O_{K}, we can tabulate indices m_{i} = [O_{K} : End(Ē_{i})]:

\begin{matrix}p & 521 & 1171 & 1741 & 2081 & 2131 & 2281 & 2591 & 3691 & 4111 & 4261 \\ m_{1} & 1 & 5 & 2 & 1 & 50 & 55 & 5 & 5 & 5 & 1 \\ m_{2} & 5 & 1 & 10 & 5 & 10 & 11 & 1 & 1 & 1 & 5 \\ p & 4391 & 5351 & 5591 & 5651 & 6011 & 6151 & 6421 & 6481 & 6521 & 6991 \\ m_{1} & 1 & 5 & 5 & 2 & 5 & 1 & 60 & 5 & 5 & 10 \\ m_{2} & 5 & 1 & 1 & 10 & 1 & 5 & 12 & 1 & 1 & 2 \end{matrix}

The primes for which (5) is inert in O_{K} are rare, and that there is no obvious preference for Ē_{1} or Ē_{2} to have the larger endomorphism ring. Can one determine a density of primes p for which (5) is inert in O_{K}?

Note that the condition [O_{K} : ℤ[π_{p}]] ≡ 0 mod 5 is equivalent, up to isomorphism, to the action of π_{p} on Ē_{i}[5] being:

π_{p} ≡ ( \begin{matrix}μ & ∗ \\ 0 & μ \end{matrix} ) mod 5.

The additional condition that [End(Ē) : ℤ[π_{p}]] ≡ 0 mod 5 is measured by the condition:

π_{p} ≡ ( \begin{matrix}μ & 0 \\ 0 & μ \end{matrix} ) mod 5.

Note that there a similar number of primes of supersingular reduction among the first 1000 primes, yet they are known to form a set of density zero.

Problem 8.1.2. Implement in SAGE an algorithm to compute End(Ē) for Ē an elliptic curve over a finite field. (Does this problem make sense for the special fiber of a Néron model as well?)

For higher dimensional modular abelian varieties, it would be interesting to have algorithms to determine the exact endomorphism rings at p, and to characterize the primes at which the reduction Ā has p-rank r in 0 ≤ r ≤ g = dim(A).

Problem 8.1.3. Let A be an abelian variety of dimension ≥ 1 attached to a newform and let p be a prime of good reduction. Find an algorithm to compute the exact endomorphism ring End(Ā/𝔽_{p}).

Problem 8.1.4. Let A be an abelian variety of dimension ≥ 1 attached to a newform. Give an algorithm to compute set of primes at which the reduction Ā/𝔽_{p} has p-rank r with 0 ≤ r ≤ g = dim(A).

Note that the endomorphism rings at ordinary primes are CM orders, and the canonical lift of the reduction Ā is a CM abelian variety. A database of invariants of CM moduli for small genus would aid in classifying these endomorphism rings (at small primes).

Problem 8.1.5. Create a database of invariants of CM moduli for small genus.

8.2 Endomorphism Rings over Number Fields

The problem of computing End(A/ℚ) has been solved by Stein (and is implemented in MAGMA). Stein has not published the algorithm, though he wrote it up.

Problem 8.2.1. Implement in SAGE Stein’s algorithm to compute End(A/ℚ).

Problem 8.2.2. Design an algorithm to compute End(A/\overline{ℚ}). This would use Ribet’s theory in [Rib92] combined with some ideas related to splitting of quaternion algebras (see [GL01]).

Problem 8.2.3. Implement in SAGE a basic framework for computing with modular abelian varieties. Stein implemented such a package in MAGMA. Much of the solution could be simply a “design document” and some basic classes.

There is also a paper of Calegari-Stein (unpublished) that goes into great detail in one case.

Problem 8.2.4. Help finish Calegari-Stein.

8.3 Component Groups

At the primes p | N of bad reduction, we study the component group Φ_{p} of the Neron model of A/ℤ_{p}. The orders of the groups finite abelian groups Φ_{p}(𝔽_{p}) are called the Tamagawa numbers c_{p}. The component group of a representative abelian variety in a class [A] is an isomorphism invariant, which can likewise vary within the isogeny class.

When p || N (exactly divides) there is an algorithm to compute #Φ_{p}(\overline{𝔽}_{p}) (see [KS00, CS01]; and it is implemented in MAGMA). This algorithm can also be used to compute the Tamagawa number c_{p} = #Φ_{p}(𝔽_{p}) up to a power of 2.

Problem 8.3.1. Find an algorithm to compute #Φ_{p}(\overline{𝔽}_{p}) when p^{2} | N.

There are standard bounds due to Oort, Lenstra, Lorenzini, etc., on the component group at primes p for which p^{2} | N. These are implemented in MAGMA.

Problem 8.3.2. Find an algorithm to compute #Φ_{p}(𝔽_{p}) when || N. I.e., remove that we currently only know how to compute #Φ_{p}(𝔽_{p}) up to a power of 2.

Problem 8.3.3. Implement in SAGE the Conrad-Kohel-Stein algorithm to compute #Φ_{p}(\overline{𝔽}_{p}).

8.4 Mordell-Weil Groups and Torsion subgroups

The Mordell-Weil group of an abelian variety (and its torsion subgroup) is again an isomorphism invariant, which varies up to finite index in the isogeny class. In the case of the above elliptic curves E_{1} and E_{2} we have

E_{1}(ℚ) ≅ ℤ/5ℤ while E_{2}(ℚ) = {O}.

Computing Mordell-Weil groups is very well studied, though still there is no provably-correct algorithm for computing it. Computing torsion subgroups for modular abelian varieties is less well studied. The paper [AS05] describes some algorithms that give upper and lower bounds.

Problem 8.4.1. Given a modular abelian variety A attached to a newform f ∈ S_{2}(Γ_{0}(N)) find an algorithm to compute #A(ℚ)_{tor}.

Problem 8.4.2. Given a modular abelian variety A attached to a newform f ∈ S_{2}(Γ_{0}(N)) find an algorithm to compute the group structure of A(ℚ)_{tor}.

8.5 Analytic invariants

Definition 8.5.1 (Modular degree). If A is an optimal quotient of J = J_{0}(N) the modular degree of A is the degree of the composite map

A^{∨} → J^{∨} ≅ J → A,

where we may identify J^{∨} with J since J is a Jacobian of a curve with a rational point.

The period lattice Λ for A can be described in terms of a pair of matrices (Ω_{1}, Ω_{2}) such that Λ = ℤ^{g} Ω_{1} + ℤ^{g} Ω_{2}. The volume of this lattice is one of the invariants which enters into the BSD Conjectures.

An analytic approach is the only known general way to compute the modular degree of an optimal quotient A of J_{0}(N). More precisely, there is a purely algebraic algorithm (which involves the theory of the analytic period lattice), which allows one to compute the modular degree. See [KS00] and the MAGMA source code. When A has dimension 1 there is an alternate algorithm due to Mark Watkins to compute the modular degree. It involves making computation of Sym^{2} L explicit and using Flach’s theorem.

Remark 8.5.2. (From Mark Watkins:)

”Flach’s Theorem” should be (maybe) ”Shimura’s formula” or something. Flach’s theorem relates L(Sym^{2}, edge) to the Bloch-Kato conjecture, whereas the Shimura work relates it (via Rankin convolution) to the modular degree (at least for curves that are not semistable, getting the fudge factors correct probably is mentioned first in Flach, but he doesn’t exactly work out the factors explicitly).

However, I think the best reference for the passage from L(Sym^{2}) to the modular degree is in Flach’s paper:

   [ ] [10] MR1300880 (95h:11053) Flach, Matthias On the degree of
   modular parametrizations. Seminaire de Theorie des Nombres, Paris,
   1991--92, 23--36, Progr. Math., 116, Birkhaeuser Boston, Boston, MA,
   1993. (Reviewer: Henri Darmon) 11G05 (11F30 11F33 11G40)

Problem 8.5.3. Is there any analogue of Watkins algorithm for any abelian varieties of dimension bigger than 1?

Remark 8.5.4. (From Mark Watkins) When I visited Barcelona I talked with Jordi Quer about a variation for ℚ-curves of Problem 8.5.3, but nothing ever happened with it. Cremona says that he has rejected a few papers that do nothing more than compute the modular degree for one or two ℚ-curves.

8.6 The Manin constant

Let A be a quotient of J_{0}(N).

Definition 8.6.1 (Manin constant). The Manin constant of A is the index in its saturation of H^{1}(A, Ω_{A/ℤ}) in S_{2}(Γ_{0}(N); ℤ) under the appropriate natural identifications (see [ARS06]).

Problem 8.6.2. Find (and implement) an algorithm to compute the prime divisors of the Manin constant of A.

This is closely related to understanding q-expansions at cusps other than ∞.

Problem 8.6.3. Give a modular form f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) compute the q^{1/h}-expansion of f at all the cusps.

Remark 8.6.4. (From John Cremona:) Prob 8.6.3 was dealt with in Delaunay’s thesis.

Problem 8.6.5. Implement in SAGE Delaunay’s algorithm to compute the q^{1/h}-expansion of a modular form at all cusps.

Problem 8.6.6. Give a modular form f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{1}(N)) compute the q^{1/h}-expansion of f at all the cusps.

8.7 Birch and Swinnerton-Dyer Conjectures

The Birch and Swinnerton-Dyer conjectures assert that certain ratios of real and integral invariants are precisely determined by the L-series of an abelian variety.

8.7.1 Refined Conjectures

In order to understand these better, one should look at not just the volume of the period lattice, and the orders of these finite groups or volumes of lattices, but the finite groups, lattices, and endomorphism rings as stuctures themselves. There are refined conjectures (due to Diamond, Flach, and others) about how these groups are related. These involve “fitting ideals”, representation theory (to make better sense of the L-function), etc.

Problem 8.7.1. Search the literature for these more refined conjectures. Write them up clearly with new examples that illustrate them.

8.7.2 Intrinsic Characterization of Optimality

There is a notion of an optimal quotient A such that J_{0}(N) → A is minimal in its isogeny class. One derives different optimal quotients of J_{0}(N), J_{1}(N), and from the Jacobians of Shimura curves. It would be very interesting to have algorithms for computing structural isomorphism invariants which distinguish these quotients. See work of Glenn Stevens [Ste89] for a conjectural answer in the case of elliptic curves (and recent work of Nike Vatsal (and Stein-Watkins [SW04]) for proofs of Stevens’ conjectures.

Remark 8.7.2. (From Mark Watkins) This only makes sense (in general) over ℚ, as else you can have that neither an isogeny or its dual is étale.

8.8 The Shafarevich-Tate Group

8.8.1 Verifying the Full Conjecture for Elliptic Curves

Stein has done substation work with many students on explicit verification of the full Birch and Swinnerton-Dyer conjecture (the exact formula) for elliptic curves over ℚ. This verification amounts to compute the exact order of the Shafarevich-Tate group � for elliptic curves.

Problem 8.8.1. Aron Lum and the Stein (mostly!) wrote a paper on explicit verification of BSD for CM elliptic curves up to some conductor. Help finish this paper and get it ready for publication. We are OK with sharing publication credit.

Problem 8.8.2. Cristian Wuthrich and Stein (mostly Wuthrich) have written a bunch of code related to using Peter Schneider’s work on p-adic analogues of the BSD conjecture to compute #� at certain primes where the methods of Kolyvagin and Kato fail. Help us write a paper and carry out the calculations (note – you won’t be a coauthor of this, but you’ll be thanked).

Remark 8.8.3. (From Christian Wuthrich.) Note that the paper mentioned above, as far as I have written it is, to my taste, more or less done. I should add some data of numerical results which you can of course ask the students to produce. But there is no need or interest for a long list. I have not written yet the introduction nor the part I named technical details (but I am not sure if I actually want to do that).

Of course, I am very happy that part (or the whole of) shark will be included in sage, on the other hand, I am not willing to add anyone to the list of authors of the paper on it.

Remark 8.8.4. (From Christian Wuthrich.) Schneider’s (and simultanoeously Perrin-Riou’s work) is strictly speaking not on the p-adic BSD. The most important result to use is Kato’s which links the algebraic to the analytic side. Look in the article we write together for a tigher bound in the case L(E, 1) is not zero. Your katobound in sage is not sharp.

8.8.2 Finiteness of �

A major very central conjecture in modern number theory is that if E is an elliptic curve over ℚ then

� (E) = ker(H^{1}(ℚ, E) → ∏ H^{1}(ℚ_{v}, E))

is finite. This is a theorem when ord_{s=1} L(E, s) ≤ 1, and is not known in a single case when ord_{s=1} L(E, s) ≥ 2. Proving finiteness of � for any curve of rank > 1 would be a massively important result that would have huge ramifications. Much work toward the Birch and Swinnerton-Dyer conjecture (of Greenberg, Skinner, Urban, Nekovar, etc.) assumes finiteness of � . Note that if L(E, 1) ≠ 0 or L′(E, 1) ≠ 0, then it is a theorem that � (E/ℚ) is finite; there isn’t even a single curve with L(E, 1) = L′(E, 1) = 0 for which finiteness of � (E) is known.

As far as I can tell nobody has even the slightest clue how to prove this. However, we can at least try to do some computations.

Problem 8.8.5. Let E be the elliptic curve defined by y^{2} + y = x^{3} + x^{2} − 2x of conductor 389. This curve has rank 2.

1. Verify that � (E)[p] is finite for 5 primes p.

2. Verify that � (E)[p] is finite for all primes p < 100.

Remark 8.8.6. (From Christian Wuthrich.) Seems doable. I quickly run shark up to p = 53, it does not take too long. As far as I remember I never actually checked if shark is optimal when computing the p-adic L-function.

Remark 8.8.7. In theory one can verify that � (E)[p] = 0 for any p using a p-descent. In practice this does not seem practical except for p = 2, 3. For p = 2

use mwrank or simon two descent in SAGE. For p = 3 use three selmer rank

in SAGE (this command just calls MAGMA and runs code of Michael Stoll).

Remark 8.8.8. The work of Cristian Wuthrich and Stein mentioned in Section 8.8.1 could be used to verify finiteness for many p. And Perrin-Riou does exactly this in the supersingular case in [PR03]. (In fact, she does much more, in that she computes � (p) in the whole ℤ_{p} tower. Shark contains now her computations with a few modifications. – from Christian Wuthrich.)

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