Hermite forms and torsion points

I spent most of the day polishing off a paper called Heuristically Fast Computation of Hermite Normal Forms of Random Integer Matrices with Clement Pernet and Robert Bradshaw. It's about a very fast algorithm Clement and I came up with (building on lots of existing work!) to compute Hermite forms. Some interesting things happened -- first, of course, we didn't completely finish the paper today. Second, Robert nailed down a very serious but "silly" efficiency bug in our current Sage implementation that Craig Citro had introduced, which made it VASTLY slower than it should be (I had stupidly positively reviewed that patch, too). Third, as with writing any paper, we had to carefully prove several key points in the algorithm that we had brushed aside before, and some of these were quite surprising to me. For example, in computing the Hermite form of a square nonsingular matrix with determinant d, one should in fact work modulo d, not 2*d as we had done before (we had blindly followed what Allan Steel says he does in Magma, which is actually overkill). We're running a bunch of benchmarks of Sage versus Magma on one of my new Sun 24-core 128GB RAM boxes, which is very fun. If our paper is sufficiently well written, Allan will referee it and the next version of Magma will be at least as fast as our implementation, since we explain all of our tricks.

I also talked with Ralph Greenberg for a while today about the field Q(E[ell^n]) where E is an elliptic curve over Q, and we assume that the ell-adic representation is surjective. We used group and Galois theory to compute all the abelian quotients of the Galois group G of Q(E[ell^n]). One deduces that the only abelian quotient is (Z/ell*Z)^*, hence the maximal abelian subfield of Q(E[ell^n]) is Q(zeta_ell). This can be used along with the Chebotarev density theory to prove that if P is a point on E(Q) of infinite order and m is an integer divisible only by primes such that the ell-adic representation is surjective, then there are infinitely many primes p such that the reduction of P modulo p has order divisible by m. And that's interesting, since it's a step in a paper I'm writing about a generalization of the Gross-Zagier formula to higher rank. There are other ways to prove the above reduction statement, I think in more generality, which I thought through last night. But I was just very curious about the structure of G, which is why Ralph and I talked it through. The basic idea for understanding it is to use a few tricks, including that PSL_2 is simple, that there is always a square root of the matrix -1 in SL_2(F_ell), and that the kernel of reduction from SL_2(Z/ell^nZ) to SL_2(F_{ell}) is isomorphic to trace zero matrices with the conjugation action, hence a simple representation.

Modular degrees, congruence moduli, and the Heegner-Kolyvagin subgroup of Mordell-Weil

Today in the advanced number theory reading seminar, Robert Bradshaw talked about the algorithm of Delaunay-Watkins for computing modular degrees using L-functions. In particular, he talked about some of the background definitions involving the symmetric square L-function and the Petersson norm. Craig Citro (in the audience) kept bringing up the adjoint L-function, since it is very relevant to his Ph.D. thesis work. There is a formula for L(Sym^2(f),2) that involves the modular degree, and there is a formula that involves L(Ad(f),1) and the congruence modulus. There's a paper of Agashe-Ribet-Stein (me) that proves that the modular degree m divides the congruence modulus c for elliptic curves, and we conjecture that 2*ord_p(c/m) <= ord_p(N), where N is the conductor of the curve. Ken Ribet in fact proved this when ord_p(N) <= 1, but none of us have made any progress when ord_p(N) >= 2. I'm very curious if rephrasing these divisibilities as relations about L-functions yields any new insight either about L-functions or the conjecture? Or nothing? I have no idea yet.

I also thought some about various ways of defining subgroups of Mordell-Weil groups of elliptic curves of arbitrary rank. The result is that now I have I think three a priori completely different definitions of a subgroup V of E(K), where K is a Heegner quadratic imaginary field, and I suspect that all three definitions give the same group. The first definition is the subgroup of E(K) that is in the kernel of all the maps to component groups and to the dual of all groups Sha(E/K)(p^oo), where dual is defined by lifting to the Selmer group and using the cup product. The second definition is the intersection of the sums of the inverse images of all subgroups of E(K_lambda)/M E(K_lambda) generated by Kolyvagins points P_lambda. The third subgroup is the inverse image of the subgroups of J_0(N*lambda)(K)/M generated by Heegner points x_K in J_0(N*lambda)(K).

Gross Zagier, Hilbert Class Polynomials, and Rank 2 Curves with 3 Sha

Today I gave a talk on the Gross-Zagier formula in the this seminar. I stated the main formula over the Hilbert class field, then explained exactly how to use it to deduce a formula over the quadratic imaginary field K. The proof involves using a summation trick and projecting onto eigencomponents of the Hecke algebra. I also talked about the relationship between L(f,chi,s), L_A(f,s), and the ranks of the eigencomponents of J(H)_C.

Right after my talk, several of us went to Microsoft Research, where we had lunch, then Drew Sutherland gave a superb talk on his very impressive multi-modular algorithm for computing Hilbert Class Polynomials modulo one cryptographic sized prime P. The main advantages of his multi-modular method are that it is very easy to parallelize, is extremely memory efficient compared to all other known approaches, and in several situations in the algorithm he can lower the constant in the time complexity by a big factor by using explicit models for X_1(N) (and some other tricks). Very nice.

When I got home, Mark Watkins had sent me a table of 630 elliptic curves of prime conductor and rank 2 that have Shafarevich-Tate groups of order divisible by 9 (see this Sage worksheet). The smallest-conductor example E we know of is y^2 + x*y = x^3 - x^2 + 94*x + 9, which has conductor "only" 53295337. The field K=QQ(sqrt(-7)) satisfies the Heegner Hypothesis, and E^K has rank 1. Also, an hour computation shows that Sha(E^K)_an = 1. I double checked that indeed Sha(E/QQ)_an = 9. Of course the torsion subgroup and Tamagawa numbers are trivial (this follows automatically from the Neumann-Setzer classification). So one expects that the Heegner-Kolyvagin subgroup [which I've never "publicly defined"] of E(K)=Z x Z x Z is a subgroup of index 3 = sqrt(#Sha). I wonder if it is possible to somehow compute -- at least conjecturally(!) -- which subgroup it is? Possibly, yes, by finding the subgroup of E(QQ)/3 E(QQ) that maps to a subgroup of Sel^(3)(E/QQ) that is orthogonal to the inverse image of Sha(E/Q)[3]. I wonder if complexity-wise such a computation is impossibly hard or reasonable?

L-functions: a seminar and the Gross-Zagier L-function

Today we had the first meeting of the Advanced Grad Student Number Theory Research Reading Seminar at UW. The participants are me, Robert Miller, Robert Bradshaw, and Craig Citro, and we intend to talk about papers of Dokchitser, Gross-Zagier, Watkins, Delauney, Kartik, Bosman, Kolyvagin, Ghate, and Hida. The unifying theme is mostly about explicit applications of L-functions and modular forms to very hard problems in algebraic number theory. We first had an organizational meeting and decided what we'll be talking about, then I gave a talk on Dokchitser's paper (basically following the talk I gave at UT Austin a few days ago, but greatly abbreviated due to lack of time).

After that, I went out to lunch with Tom Boothby, Sourav San Gupta, Craig Citro, and Robert Bradshaw at Cedars, where we talked about a reading course Boothby and Gupta are doing on algebraic number theory (basically, exercises from Marcus's book).

Next, Robert Bradshaw, Craig Citro, and I sat down in my office and figured out precisely how to use Dokchitser's algorithm to compute the Gross-Zagier L-function L_A(f,s). This involved using the Legendre duplication formula (page 217 of Ahlfors), explicitly computing the representation numbers r_A(n) using quadratic forms (Robert Bradshaw has a super-fast Cython implementation of this which is 20 times faster than Magma's, which uses careful bounds on the ellipse one enumerates over), and explicit computation of the product of two Dirichlet series as a Dirichlet series, which again Robert coded up very quickly by noting that one of the series is very sparse. Then we got all this to work via the Sage/Dokchitser interface, and it computed the power series of L_A(f,s) at 1 for 37a and D=-40. Robert will be making this all much more systematic as a patch to Sage, and extend it to work for any newform of any weight or level (for Gamma0(N)), so that we can do many systematic computations. In particular, I'm very interested in what happens when ord_{s=1} L_A(f,s) >= 3 (it's always odd), and also what happens when f isn't attached to an elliptic curve.

Dokchitser's Algorithm for Computing L-functions

On the trip Thursday from Seattle to Austin, I read Tim Dokchiter's paper "Computing special values of motivic L-functions". This is Tim's oldest paper listed here, and some of his other papers build on it. It describes a very general algorithm for computing "anything" numerical about motivic L-series, which are Dirichlet series with meromorphic continuation, functional equation, etc. We are implemented this algorithm in Sage right now, since it's potentially very important to many things I care about:

1. Birch and Swinnerton-Dyer, Bloch-Kato, Stark Conjectures
   
2. Heegner heights: Gross-Zagier + Zhang; computation of twisted L-series by characters of class groups, and also computing Rankin-Selberg convolutions.
   
3. Computing conductors of curves, local factors at primes of bad reduction
   
4. Computing the Petersson pairing by computing special values of symmetric square L-functions; get explicit Poincares series from this. Also, the related problem of
   computing modular degrees of modular abelian varieties and modular motives analytically.
   

Dokchitser's algorithm is already implemented (by him) in GP/PARI and Magma, but a highly optimized implementation done directly in Sage would be extremely desirable. Nick Alexander, Jen Balakrishnan, and Sourev San Gupta have all worked on this but their implementation is still rough and has bit rotted. We already use the GP/PARI version in Sage a lot, but the main motivation for doing a new implementation is that it will be more flexible wrt the above problems, and we can potentially optimize it.

Speaking of speed, I tested Pari versus Magma's Dokchitser implementation for the elliptic curve [1,2,3,4,5], and Pari was 2 to 6 times faster on everything I tried for that L-series. Somebody conjectured this might be that Dokchitser knows pari better than Magma, or that Pari has more relevant low level data structures that he could use.


Today I gave a talk on Dokchitser's algorithm, focused on explaining the key "theoretical" ideas of the algorithm, without getting bogged down in details. In my talk, I discussed the applications of Dokchitser's algorithm, then wrote down his formula for L^*(s) in terms of G_s(t) and the a_n, and emphasized that the G_s(t) depend only on the gamma factor. Then I said some words about how to compute G_s(t), and finally I gave almost all the gory details of derivation for the formula for L^*(s) in terms of G_s(t) and a_n. I was surprised to end up giving all the details. Mike Rubinstein and Fernando Rodriguez-Villegas both made a lot of helpful comments during they talk.

Gross-Zagier: Heegner Points and Derivatives of L-series

Today, after answering at least a dozen emails from new users about basic design decisions in Sage, I read chapters II and V of Gross and Zagier's seminal paper "Heegner Points and Derivatives of L-series" (at the Vivace coffee shop by REI in downtown Seattle). Chapter II is a very technical derivation of a formula for local archimedian heights of pairings of certain Heegner points. It mainly involves a careful construction of a certain "resolvent kernel function" G by a limiting process, deriving relations between that function and the local archimedian contributions to the height, and dealing with various cases. The function G looks perhaps computable, though I've never heard of anybody computing it; there are some related examples in the last chapter of this paper.

I only skimmed chapters III and IV, which I'm saving for later, since they're 50 pages of very technical arguments.

Chapter V is extremely exciting, since it ties together the ideas from the previous chapters to finally relate heights of Heegner points to L-values. The basic idea is that using Rankin's method, Gross and Zagier write down a "horrendous" formula for the coefficients of a certain modular form that encodes the linear functional that sends a newform f to the value of the derivative at 1 of a certain L-series. They also use that G function mentioned above and explicit computation with local heights to write down another horrendous formulas for height pairings of Heegner points. They then observe in Chapter V that the formulas as the same! (And, yes, they include an excited exclamation point at that point in the paper.) The rest of Chapter V involves deducing the statements that originally motivated Gross's interest in this problem (e.g., the conjectures of Birch related to the BSD conjecture), an application to weight 1, and some ruminations on higher weight analogues that involve Grothendieck motives attached to higher weight modular forms, though Gross-Zagier talk only of Deligne's cohomology theory, since this paper was written before Scholl's paper on motives attached to modular forms. Much of these later deductions will be very fun to generalize to higher derivatives, since they're basically formal applications of their main formula.

Idea/Question: Something I've long wondered about is whether there is a way to compute the *conjectural* order of Sha and the regulator for modular abelian varieties A over the rational numbers with positive rank. Reading Chapter V of Gross-Zagier makes me think there is in the common case when rank(A) = dim(A), which would be incredibly useful for making a large table that generalizes Cremona's book to dimension bigger than 1. Here's the very roughly sketched idea. Given a newform f of degree > 1 over the rational numbers, there is a corresponding abelian variety A=A_f of dimension d > 1. It is fairly easy to compute L(A,s), and even the leading coefficient L^(d)(A,1) and BSD volume Omega_A. Assuming A has rank d, according to the BSD conjecture, L^(d)(A,1)/Omega_A is a nonzero rational multiple alpha of the regulator. I wonder if one can get any information at all -- even conjecturally -- about this multiple by computing something explicit involving the various constructions in the Gross-Zagier paper? Some sort of calculation using Gross-Zagier should give the regulator of the subgroup generated by the Heegner point and its Galois conjugates, and this will be a square multiple of the true regulator. Comparing that with what BSD gives for Reg*Sha, and throwing in knowledge of (or bounds on) the Tamagawa numbers and torsion, and an analogue of the conjecture in Section 5 that the index of the Heegner subgroup in the full group of rational points is a formula involving the square root of something involving the torsion, Tamagawa numbers, Manin constant, and order of Sha, should yield a conjectural formula for Sha. This would take working out stuff not given explicitly in Gross-Zagier in higher dimension, but I think would be do-able. This will definitely require explicitly computing something about p-divisibility or indexes of Heegner points in A(K), possibly using ideas from Johan Bosman's Ph.D. thesis.

I talked more with Robert Bradshaw about all the above, and our conclusion was a first interesting problem is just to figure out precisely what replaces the elliptic curve index [E(K):Z y_K] when E is replaced by A=A_f. Maybe [A_f(K) : T y_K] where T is the Hecke algebra? But then sqrt(#Sha(E))*prod c_p * Torsion(E) has to be replaced by something else since #Sha(A) might not be a perfect square.

Annals, Crypto, and Modular Forms

This morning I read Iwasawa's original paper where he introduces p-adic L-series of Dirichlet characters. I basically read this on a whim, since I saw it in a friend's collection of scans.

I also skimmed through all the number theory related papers published by Annals of Math during the last few years. Probably the most interesting were Kartik Prasanna's paper on integrality of Petersson norms (vol. 163, 2006, 901-967), in which he remarks "I am grateful to my advisor Andrew Wiles for suggesting the problem mentioned above and for his guidance and encouragement. The idea that one could use Iwasawa theory to prove the integrality of the Rankin-Selberg L-value is due to him and after his oral explanation I merely had to work out the details." Very humble. Another very nice paper is the fourth in Manjul Bhargava's sequence of papers about parametrizing quintic rings. There's also a nice paper by H. Darmon about his p-adic version of Heegner points.

After that I read through half my paper on generalizing the Gross-Zagier formula, and marked it up with comments.

Then I met with some "Dutch crypto guys" from Microsoft Research and had lunch. After that I went to Clement Pernet's class, where he explained Weiedemann's algorithm for computing minimal polynomials of pairs (A,v) where A is a sparse matrix and v is a vector.

Next I went to the new Number Theory and Computation seminar that Dan Shumow and I started, and saw a nice talk by Dustin Moody on how pairings provide new techniques for algebraic crypto. Next Craig Citro gave an exciting and erudite talk on enumeration of mod p modular forms, which develops in great details some ideas I discussed with him at Sage Day 7 at IPAM. I gave Craig a hard time during his talk though :-).

Now I'm dressed up as a skateboarder for Halloween.

Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves

I spent last night and this morning making changes to my paper Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves. I had been sitting on the referee report for several months, since there are four co-authors, and I naively hoped one of them would fix all the problems.

This is a paper I wrote nearly four years ago really as a summer student project: SAGE -- Summer of Arithmetic Geometry Experience -- that I ran at Harvard for a group of undergrads. Five undergrads there all independently requested to work with me on research over the summer, and this paper was one of the results of that summer.

This paper was also the first application of the Sage Mathematics Software. I tried to repeat some calculations from the paper when making improvements suggested by the referee, and was scared that I would find bugs that prevent me from doing them. Indeed, within a few seconds I was disappointed to find a showstopper bug, which fortunately I was able to fix in a few minutes.

The referee remarks about the paper were almost all very useful; they definitely improve the quality of the paper a lot.

The paper itself proves most of the BSD conjectural formula for elliptic curves of conductor up to 1000. The obvious next steps with that paper done are:

1. Mostly prove BSD for far more curves (say up to conductor 130,000). This would be subtle and interesting, since all kinds of problems that don't quite happen for conductor up to 1000 would certainly happen here.
   
2. Completely prove BSD for all curves up to conductor 1,000, except the 18 curves of rank 2. This will rely on my joint paper with Wuthrich, which will make this almost completely straightforward, except perhaps in a handful of surprising cases.
   
3. Prove the prediction of BSD about Sha at a bunch of primes for curves of rank 2. This can also be done using my joint paper with Wuthrich.
   

A Reading Seminar, Schemes, and Petersson norms

Today I had lunch with Craig Citro, Robert Miller, Robert Bradshaw, and Dan Shumow, and we mainly discussed starting an "advanced number theory reading seminar". Craig's the main impetus behind it, and it will be mainly about talking about papers like Gross-Zagier, etc.

After that I met with Tom Boothby and Sourav San Gupta and defined schemes for them, and explained how to think of maps between rings of integers as morphisms of schemes, and factorization of primes in integer rings as computation of fibers. We also explicitly computed the morphism from y=x^2 to y=0 in terms of maps on affine rings.

After that, I met with Robert Bradshaw (one of my Ph.D. students), and looked at a bunch of papers that are related to computing Petersson norms and special values of symmetric power L-functions. The problem we're looking at currently is: efficiently compute L(Sym^2(f), 2) for general newform f, and compute the Petersson norm of f. This is "essentially done," as explained in (Delaunay, 2003), when the level of f is square free. When the level is not square free, things are more interesting, and we discussed at least two ideas for dealing with this more general case: (1) try all possibilities for the bad factors and see which leads to a consistent functional equation, or (2) compute the modular degree via modular symbols and linear algebra, and use a formula as in Cremona or Zagier's papers on modular degree. We also looked at an old paper of Rankin that Kevin Buzzard sent us that computes Petersson inner products for level 1 forms -- it is horrendous, only level 1, and doesn't use symmetric square L-functions at all, and is clearly at least O(N^3) complexity (or worse).

When reading through these papers, we ran into numerous cases where we couldn't get relevant references because scans are not available online (or cost $30/each, and our university subscription doesn't go back far enough), and I don't have a scan or photocopy, and of course the library is closed. It's extremely annoying how frustrating the situation still still is regarding having access to the literature in number theory.