paul Monsky

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Apr
2
answered The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
Feb
11
revised The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
A conjectural formula for the highest degree part of H_N is given.
Jan
24
revised Questions (related to deformation theory?) about modular ideals in mod ell Hecke algebras
References given to answers to some of my questions. An error corrected.
Jan
21
revised Questions (related to deformation theory?) about modular ideals in mod ell Hecke algebras
Empirical results in level 1 and characteristics 5,7 and 13 are given. Typo corrected.
Jan
16
comment The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
Just so, Anna. And here's how I got the level 2 characteristic 11 equation for delta. j*delta=(E_4)^3, while (j-1728)*delta=(E_6)^2. So ((j*j*(j-1728)^3)*delta^5 =(E_10)^6, which is 1 mod 11. Combine this equation, the corresponding equation with j(z) and delta(z) replaced by j(2z) and delta(2z), and the characteristic 11 equation linking j(z) and j(2z), and eliminating j(z) and j(2z) you get, mod 11, a degree 30 equation linking (delta(z))^5 and (delta(2z))^5, which then gives the relation I mentioned. Experiment convinced me that it's irreducible.
Jan
13
awarded  Nice Question
Jan
13
revised What's known about the mod 2 reduction of the level l Jacobi modular equation?
Errors corrected.
Jan
13
revised What's known about the mod 2 reduction of the level l Jacobi modular equation?
added 2642 characters in body
Jan
11
revised The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
Connection with earlier questions established and a more geometric setting invoked. Results established when the characteristic is 2. 2.
Jan
9
revised The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
A comment by Anna M. is expanded into a partial answer.
Jan
9
comment The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
Some more detail on the last point at level 2. By general theory, the classical modular polynomial for j of level 2 is x^3+y^3+a+b*(x+y)+(cx^2+dxy+cy^2)+e*(xy^2+yx^2)-(xy)^2, with integer a,b,c,d and e. In fact, a=-157464000000000, b=8748000000, c=40773375, d=-162000 and e=1488. So what I'm asserting about the level 2 equation for delta when l=3 or 5 is that modulo ell, a=b=c=d=0 and e=3.
Jan
9
comment The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
... when delta=1/j, the highest degree part of the modular equation for delta corresponds to the lowest degree part of the modular equation for j. So one needs to determine various low order terms in this last modular equation, which is a delicate business.
Jan
9
comment The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
@Anna--It's trickier than you think. What you say in your last comment is wrong; delta(Nz) is transcendental over Q(delta(z)) and the coefficients in your product are not polynomials in delta(z). When ell=11 for example the irreducible relation between delta(z) and delta (2z) has degree 150 and the highest degree part is (xy)^75, But when Z/ell(delta) =Z/ell(j), as happens when ell is 2,3,5,7 or 13, then what you say is right. As for your earlier comment, the level p modular equation for j has total degree 2p and highest degree part (xy)^p and (to be continued)...
Jan
6
asked The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
Jan
1
comment Explicit generators of maximal ideals in completed Hecke algebras
One may ask analogous questions when ell> 3. But now M isn't closed under + and one works with (ell-1)/2 spaces M(w) with w even in [0,ell-3]. (M(w) is now the set of reductions of expansions of modular forms whose weight j is w mod (ell-1). Since for fixed n the mod ell reduction of n^(j-1) only depends on j mod (ell-1), the Hecke operators T_n are well defined at the characteristic ell level, and one gets a shallow Hecke algebra HE(w) acting on M(w).) But I haven't looked at this more general situation yet.
Jan
1
comment Explicit generators of maximal ideals in completed Hecke algebras
I was imprecise in defining O as the "m-completion of HE". The m-completion in the usual sense, (HE)^, the inverse limit of the HE/(m^j) is a power series ring in infinitely many variables. Instead let M(m) be the subspace of M consisting of u annihilated by a power of m. Then (HE)^ acts on M(m), and O is the quotient of (HE)^ by the ideal of elements annihilating M(m).
Dec
27
asked Explicit generators of maximal ideals in completed Hecke algebras
Dec
12
awarded  Popular Question
Dec
4
comment Higher level analogs of Nicolas-Serre theory
Computer calculations (thanks, Ira Gessel) show that when [a,b]=D^k with k<500, k=1 mod 6, then my conjectured results about the effect of T_7 and T_13 on [a,b] are true. So trying to construct these long and technical proofs is not a misguided enterprise.
Dec
2
answered Higher level analogs of Nicolas-Serre theory