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It's possible to give a semester course in number theory, free from overt algebraic geometry, that handles the zeta functions of curves over finite fields, proving the functional equation and Weil's R …

Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z] is an irreducible cubic form defining a plane curve C with a node. A lot is known about sheaves on C; for exam …

In my question I remarked that when l=2m+1 a modular form argument might be used to show that the field generated by [1], ... ,[m] over Z/2 has transcendence degree 1. Below I give an elementary proof …

Felipe Voloch referred me to two 1959 papers of Igusa in v. 81 of Amer. J. of Math. pages 453-475 and 561-577. Results from these papers and techniques I've developed on MO give an answer to my questi …

Suppose $\ell=2m+1$, $m>0$. Define $[i]$ in $\mathbb{Z}/2\mathbb{Z}[[x]]$ to be $$\sum_{n\equiv i\mod l} x^{n^2}.$$ Note that $[0]=1$, and that $[i]=[j]$ whenever $\ell$ divides $i+j$ or $i-j$. Now l …

There should be many examples where the interior vertex lies on the perpendicular bisector of one of the sides. It's best to redefine a Heronian triangle to be one with rational sides and rational are …

Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows: The subring, S, is generated b …

In the first version of this answer I gave a (necessarily incorrect) proof of the false statement that when the prime,l, is 11, then G is not in the field generated over Z/2 by the [j]. In the second …

I suppose it's bad form to answer one's own MO question, but I now have an almost complete solution to this one. I can prove: 1.----H is always in the ring S generated by the [j]. 2.----The same hol …