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visits | member for | 4 years, 3 months |
seen | 2 hours ago | |
stats | profile views | 1,999 |
Jul 2 |
awarded | Curious |
Jun 3 |
revised |
The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
unnecessary edit removed |
Jun 3 |
comment |
The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
The edit to my answer was entirely unnecessary, (and I'll get rid of it); the original argument given in the edit to the question was fine. The point that I should have gathered from Anna M. is that the kth elementary symmetric function in the delta((z+i)/p) and (p^12)*delta(pz) is a weight 12k modular form of level 1. And then of course, for our particular ell, the reduction of its expansion is a polynomial of degree at most k in F. |
May 25 |
revised |
The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
typo corrected |
May 25 |
revised |
The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13
I've fixed a gap in a proof contained in my edits to the question. |
May 20 |
awarded | Yearling |
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 |