# paul Monsky

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 Name paul Monsky Member for 3 years Seen 5 hours ago Website Location Age
 May10 revised A subring of the Serre Swinnerton -Dyer ring of level N modular power seriesTypos corrected; an e-mail answer acknowledged. Apr21 revised Solutions to $\binom{n}{5} = 2 \binom{m}{5}$ Remarks on rational solutions to the equation added. Apr20 awarded ● Nice Answer Apr19 answered Solutions to $\binom{n}{5} = 2 \binom{m}{5}$ Apr17 asked A subring of the Serre Swinnerton -Dyer ring of level N modular power series Mar31 accepted Computing certain class numbers modulo 4 Mar29 comment Computing certain class numbers modulo 4@Sarah--In the answer to your earlier question I could have used the seventh power of g= x+x^4+x^9+x^16+... rather than the eleventh. Then it would turn out that the coefficient of x^pq in the expansion is odd or even according as the class number of Q(root(-2pq)) is 4 mod 8 or 0 mod 8. And the Gauss theory of forms axx+bxy+cyy with b^2-4ac=-8pq would show that the class number is 4 mod 8 when (q/p)=-1 and 0 mod 8 when (q/p)=1. So my argument would show that g^7 isn't the reduction of the expansion at infinity of any modular form for any gamma_0. Mar29 revised Computing certain class numbers modulo 4error corrected Mar29 answered Computing certain class numbers modulo 4 Mar27 comment Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? Alternatively, if p=5 mod 8 and q=7 mod 8 then the number of ways of writing pq as s_1+2s_2+8s_3 with the s_i squares is the same as the number of ways of writing pq as s_1+2s_2+8s_3 with the s_i non-zero squares. So the coefficients of x^pq in g^11 and in (1+g)^11 are the same. Mar27 comment Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? @Sarah. That's right. Another way to say it--instead of using (1/2)*(phi-(E_4)^m) in my edited comment, use (1/2)*(phi+(E_4)^m). Mar26 accepted Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? Mar25 comment Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? Here's the argument that if u is the mod 2 reduction of an element of Z[[x]] that is the expansion at infinity of some modular form phi of weight w for gamma_0 (N) then the space spanned by the image of u under the formal Hecke operators "T_q" ,q prime, has finite dimension. For fixed N and w the Z-module of such elements of Z[[x]] has finite rank and is stable under the Hecke T_q where (N,q)=1. So the image of this module under mod 2 reduction contains u and has finite Z/2 dimension. The T_q with (2N,q) reduce to "T_q". These "T_q" stabilize the image. And only finitely many q divide N. Mar25 revised Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? Complete answer given. Mar23 awarded ● Necromancer Mar22 comment Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? @Will-My proof? is more or less a version of this idea, as I think my conjectures when the level is odd should follow from results of Igusa on the modular curve. But I have no feeling for what happens when the level is even. (I took Katz and Mazur out of the library last year, but it went unread--it's not for amateurs). Mar21 revised Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? comment incorporated into answer, slight elaboration Mar21 answered Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? Mar19 comment Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? Maybe the following approach might be helpful.(I'm guessing that the answer to your question is no.) If there is such a theta, raising it to an appropriate odd power, multiplying what you get by 1+2(x+x^4+x^9+...), subtracting off an Eisenstein series and dividing by 2 would give a modular form of integral weight whose mod 2 reduction, g, is x+x^4+x^9+.... Then a theorem of Serre would imply that for any k almost all the coefficients of g^k are 0. I wonder what the evidence is for or against this claim about g. Mar19 revised Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?typo corrected Mar19 revised Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?More observations/discoveries made. Mar11 comment Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?@Alberto: Thanks. I followed your suggestion. Mar11 revised Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?Made title more descriptive Mar11 asked Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems? Mar5 comment Level p characteristic 2 modular forms and thetasThere's a typo in my treatment of p=19. I should have written y^2+y=x^3, not y^2+y=x^3+x. Mar4 revised Level p characteristic 2 modular forms and thetasmany examples given confirming the conjecture Mar4 revised Level p characteristic 2 modular forms and thetasFurther conjectures made, and a promise to provide explicit results for p<20. Feb17 awarded ● Nice Answer Feb11 asked Level p characteristic 2 modular forms and thetas Jan6 comment Can every curve be written as $f(x)=g(y)$?Suppose you restrict attention to curves defined over Q? Is the result still true, and can you exhibit an example? What about the Klein quartic for example? I believe its Jacobian is a factor of the Jacobian of the Fermat curve of degree 7, but it's not clear to me whether it is a model of some f(x)=g(y). Jan4 answered Elementary examples of the Weil conjectures Jan3 comment Elementary examples of the Weil conjecturesBut the most elementary proof of RH for curves is that of Bombieri, which uses nothing more than RR for curves, and avoids any higher dimensional algebraic geometry. Jan1 revised Are there Heronian triangles that can be decomposed into three smaller ones?Another related construction given. A modification of the problem suggested. Jan1 answered Are there Heronian triangles that can be decomposed into three smaller ones? Dec14 comment The mod 3 reduction of some powers of deltaRamanujan determined the mod 27 reduction of the Fourier expansion of delta, and perhaps others have worked on my particular powers of delta. Dec14 comment The mod 3 reduction of some powers of delta@Will--I agree. Let g in Z/2[[x]] be the characteristic 2 analogue of f. Joel Bellaiche proved a conjecture of Nicolas and Serre, and using this found just which linear combinations of the g^k with k odd corresponded to abelian Galois representations. (In particular the g^k with k=3,5,7,19 and 21 are "abelian"--I write a little about this on other MO questions). I've experimentally confirmed a characteristic 3 analogue of Joel's result on linear combinations, but have no proofs. However my question about f^k where k=2,4,5 or 10 perhaps admits a more elementary answer--(to be continued) Dec13 awarded ● Nice Question Dec13 revised The mod 3 reduction of some powers of deltaA proof of one of my experimental results is given.