Some 2-adic Computations

Robert Coleman  William Stein

July 18, 2001

This document records email between myself and Robert.

Contents

1 The Problem 1

2 I Did the Computation 2 2.1 The MAGMA program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 A Remark about Emerton’s Thesis 5

4 Emerton’s Form: More details 6

1 The Problem

Set p = 2 and Λ^{∗} := Z_{2}[[1 + 4Z_{2}]]. For any integer m ≥ 1, let W_{2,m}^{∗} be the set of imaginary quadratic integers (i.e., they satisfy a quadratic monic polynomial with coefficients in Z) in Z_{2} that are congruent to 1 mod 4 and have norm 2^{m}. I think that

W_{2,m}^{∗} = \{γ = \frac{−a − \sqrt{a^{2} − 4 · 2^{m}}}{2} : a^{2} < 4 · 2^{m} and a ≡ −1 (mod 4)\} ,

where we choose this square root: \sqrt{a^{2} − 4 · 2^{m}} ≡ a (mod 8).

For γ ∈ W_{2,m}^{∗}, let O_{γ} be the set of orders in the ring of integers of Q(γ) that contain γ. The set O_{γ} can be computed by listing the discriminants of its elements. You can find

the γ by looking at these numbers. Suppose N^{2}D = a^{2} − 4 · 2^{m} where D is square free. Then the orders containing γ = (a + N\sqrt{D})/2 correspond to the divisors of N.

Let

B_{m} = ∑_{γ∈W_{2,m}^{∗}} ∑_{O∈O_{γ}} h(O) · \frac{[γ]}{γ^{2} − 2^{m}} .

Here for γ ∈ 1+4Z_{2}, [γ] is the element in the group part of Λ^{∗}, so the above could be viewed as an expression of B_{m} as a Z_{2}-linear combination on the formal vector space generated by the elements of the additive group of 1 + 4Z_{2}.

Let Q(T) be the series in Λ^{∗}[[T]] such that

T d log Q(T) = ∑_{m≥1} B_{m}T^{m}.

Then

Q(T) = exp ( ∑_{m≥1} \frac{B_{m}}{m} T^{m}) .

If X = [5] − 1, then we can identify any element of Λ^{∗} as a series in X. Explicitly,

[γ] = (1 + X)^{log γ/ log 5}.

It then makes sense to consider the reduction modulo 2 of Q(T) as a power series ∑ a_{n}(X)T^{n} over F_{2}[[X]]. I (RC) conjecture that ord_{X}a_{n}(X) = \binom{n}{2}. I (RC) checked this for n ≤ 4.

At present, I don’t even know if a_{n}(X) ≠ 0 for large n.

2 I Did the Computation

I coded and ran the mod-2 characteristic series computation that you suggested. The mod-2 result, up to O(T^{10}) is given below. The valuation of the coefficient of T^{n} is n choose 2, just as you conjecture. Let me know if the data looks incorrect in any way.

1 +
  (1 + X + X^2 + X^3 + X^16 + X^17 + X^18 + X^19 + X^32 + X^33 + X^34 + X^35
+ X^48 + X^49)*T +
  (X + X^2 + X^5 + X^6 + X^8 + X^9 + X^12 + X^13 + X^16 + X^17 + X^20 + X^21
+ X^24 + X^25 + X^28 + X^29 + X^32 + X^34 + X^36 + X^38)*T^2 +
  (X^3 + X^5 + X^7 + X^8 + X^11 + X^13 + X^15 + X^16 + X^17 + X^18 + X^20 +
X^22 + X^25 + X^26 + X^28 + X^30 + X^32 + X^33 + X^34 + X^35 + X^36 + X^37 +
X^38 + X^39 + X^40 + X^41 + X^42 + X^43 + X^44 + X^45 + X^46 + X^47)*T^3 +
  (X^6 + X^7 + X^9 + X^10 + X^12 + X^13 + X^16 + X^19 + X^20 + X^21 + X^22 +
X^23 + X^24 + X^26 + X^34 + X^35 + X^38 + X^39 + X^41 + X^42 + X^44 + X^45 +
X^49)*T^4 +
  (X^10 + X^11 + X^13 + X^16 + X^18 + X^19 + X^21 + X^22 + X^24 + X^25 + X^27
+ X^33 + X^37 + X^38 + X^40 + X^43 + X^44 + X^45 + X^49)*T^5 +
  (X^15 + X^16 + X^17 + X^18 + X^19 + X^20 + X^22 + X^25 + X^26 + X^27 + X^28
+ X^30 + X^31 + X^32 + X^33 + X^36 + X^37 + X^38 + X^43 + X^44 + X^45 + X^47
+ X^48 + X^49)*T^6 +
  (X^21 + X^23 + X^24 + X^26 + X^30 + X^33 + X^40 + X^42 + X^47 + X^48 +

X^49)*T^7 +
  (X^28 + X^29 + X^30 + X^31 + X^33 + X^35 + X^38 + X^40 + X^41 + X^43 + X^45
+ X^46 + X^47)*T^8 +
  (X^36 + X^37 + X^41 + X^42 + X^46 + X^49)*T^9
  + ...

2.1 The MAGMA program

For the record, here’s the MAGMA program I used to produce the above data. It doesn’t rely on any special packages. To use it, save it to a file named ”code.m”, then type

  > Attach("comp.m");
  > time Bm := [B_m(m) : m in [1..9]];
  > time f := QT(Bm);
  > R<q> := Parent(f);
  > QTmod2(f+O(q^10));
// some MAGMA code by William Stein.
/* To run, type
 > Attach("comp.m");
  > time Bm := [B_m(m) : m in [1..9]];
  > time f := QT(Bm);
  > R<q> := Parent(f);
  > QTmod2(f+O(q^10));
*/
p_prec := 100;
s_prec := 50;
Q2<pi> := pAdicField(2,p_prec);
R<X> := LaurentSeriesRing(Q2);
Rmod2<Y> := LaurentSeriesRing(GF(2));
intrinsic Wmstar(m::RngIntElt) -> SetEnum
{}
   b := Floor(Sqrt(4*2^m));
   if IsEven(m) then
      b := b-1;
   end if;
   W := { a : a in [-b..b] | a mod 4 eq 3 };   // could be made faster!
   return W;
end intrinsic;

function class_number_sum(a, m, b)
assert Type(a) eq RngIntElt;
assert Type(m) eq RngIntElt;
assert Type(b) eq RngIntElt;
D, N := SquareFree(b);   // a^2-4*2^m = D*N^2, with D squarefree.
return &+[ClassNumber(D*n^2) : n in Divisors(N)];
end function;

function bracket_gamma(a, m, b)
assert Type(a) eq RngIntElt;
assert Type(m) eq RngIntElt;
assert b lt 0;
root := Sqrt(Q2!b);
if Integers()!root mod 4 ne a mod 4 then
   root := -root;
end if;
assert Integers()!root mod 4 eq  a mod 4;
gamma := (-a - root)/2;
return (1+X+O(X^s_prec))^(Log(gamma) / Log(1+pi)), gamma;
end function;

intrinsic Sum_over_O_gamma(a::RngIntElt, m::RngIntElt) -> FldLocElt
{}
b := a^2 - 4*2^m;
bg, gamma := bracket_gamma(a, m, b);
cn := class_number_sum(a, m, b);
return cn * bg / (gamma^2 - 2^m);
end intrinsic;

intrinsic B_m(m::RngIntElt) -> FldLocElt
{}
return &+[Sum_over_O_gamma(a, m) : a in Wmstar(m)];
end intrinsic;
intrinsic QT(Bm::SeqEnum) -> RngSerLaurElt
{}

   S<T> := LaurentSeriesRing(R);
   f := &+[(Bm[m]/m)*T^m : m in [1..#Bm]];
   return Exp(f);
end intrinsic;
function ReduceMod2(a)
   if Type(a) eq FldLocElt then
      return Integers()!a mod 2;
   end if;
   if Type(a) eq RngSerLaurElt then
     seq, n := Eltseq(a);
     seqmod2 := ReduceMod2(seq);
     return Rmod2!seqmod2 * Rmod2.1^n;
   end if;
   if Type(a) eq SeqEnum then
     return [ReduceMod2(b) : b in a];
   end if;
end function;
intrinsic QTmod2(Q::RngSerLaurElt) -> RngSerPowElt
{}
   S<T> := PowerSeriesRing(Rmod2);
   return &+[ReduceMod2(Coefficient(Q,n))*T^n :
                     n in [0..AbsolutePrecision(Q)-1]];
end intrinsic;
/*
PHRASEBOOK:
time Bm := [B_m(m) : m in [1..10]];
time Append(~Bm,B_m(#Bm+1));
time f := QT(Bm);
R<q> := Parent(f);
QTmod2(f+O(q^13));
*/

3 A Remark about Emerton’s Thesis

(I’m not sure what this is a remark about, *exactly*.)

I know that Q(T) is entire and it follows from Emerton’s work that it has one zero of valuation −1. (This means, in particular, that v(a_{1}) = 1 and v(a_{n}) > n.) Using this, he constructed a family of eigenforms

f(X)(q) = ∑_{n≥1} b_{n}(X)q^{n}

where the b_{n}(X) are analytic functions bounded by 1 on 1 > |X| > |8| and f(X) has weight-character determined by

5 ↦ (1 + X), −1 ↦ 1,

so is classical of weight k if 1 + X = ζ5^{k−1} for k ≥ 2 and ζ is a root of unity.

There seems to be some confusion (at least in my mind) whether a level one weight k modular form corresponds to the character γ ↦ γ^{k} or γ ↦ τ^{k}(γ)γ^{k−1}.

Also, v(a_{2}(X)) = v(X). (See the main theorem in Chapter 4, only I have to translate the notation. In particular, u = a_{2}(X), t = 5^{14}(X + 1) − 1.)

Now one can also reduce the b_{n}(X) modulo 2 and get elements in F_{2}[[X]]. Let F (X) be the reduction of f(X) mod 2. It follows from Proposition 5.8 that

F (X)(q) ≡ q + Xq + Xq^{2} + Xq^{3} + Xq^{5} mod (X^{2}, q).

It would be nice to know something mod X^{3}. I don’t believe that b_{9}(X) mod 2 is divisible by X (because Δ ≡ f(5^{11}) mod 8).

4 Emerton’s Form: More details

What I want to understand are the elements of F_{2}[[X]] that the b_{n}(X) reduce to. The Eisenstein family E(q) is defined as follows. For n ≥ 1 set

σ^{∗}(n) := ∑_{d|n, (d,p)=1}[d]d^{−1},

as an element of Z_{2}[[X]] ≅ Z_{2}[[1 + 4Z_{2}]]. Again, we identify X with [5] − 1. Also, in the definition of σ^{∗}(n), set p = 2. Let ζ^{∗} be the 2-adic L-family, which we view as an element of X^{−1}Z_{2}[[X]]. (I’ll be clearer about what this is later.) Then,

E(q) = 1 + \frac{2}{ζ^{∗}} · ∑_{n=1}^{∞} σ^{∗}(n)q^{n},

Let C(q) ∈ B := F_{2}[[X]][[q]] be the reduction modulo 2 of E(q)/E(q^{2}). Let j_{2} be the function in Emerton’s thesis, and set

Z = 1/j_{2} = q ∏(1 + q^{n}).

Let U be the operator on B defined by

∑ a_{n}q^{n} → ∑ a_{2n}q^{n},

and let S be the map on qB[X^{−1}] → qB[X^{−1}] given by

f(q) → U(C(q)f(q))/X.

Then S^{i}(Z(q)) ∈ qB (by page 49 of Emerton’s thesis). Also to get started we may only want to look at the first 64 coefficients of Z(q) and C(q). Now let Z_{n}(q) = S^{2^{n}}(Z(q)). Then Z_{n}(q) should converge to the q-expansion of a multiple G(q) of F (q)/E(q). In fact, I am hoping

Z_{n}(q) ≡ G(q) mod X^{2^{n}}.

Of course, if we only look at the first 2^{m} terms of Z(q) and C(q) we will only get the first 2^{m−n} terms of Z_{n}(q). If all this works, if m = 64, after 4 iterations, we should get the first four coeficients of F (q) modulo X^{4}.