Suppose that $p$ is a prime, $A$ is a subset of $\mathbb F_p$, and $P$ is a polynomial over $\mathbb F_p$ of degree $d$. Using Weil's bound, it is not difficult to show that $$ \left| \sum_{a,b\in A} \left(\frac{P(a+b)}{p}\right)\right| \le d|A|\sqrt{p} $$ (where $(\cdot/p)$ is the Legendre sym...

I am looking for references that discuss Hecke operators $T_n$ acting on modular forms for the principal congruence subgroup $\Gamma(N)$ of the modular group $SL(2,Z)$ and am happy to restrict to the case that $(n,N)=1$. Most textbooks (Diamond and Shurman, Koblitz etc.) that discuss Hecke operat...

Question. Assume the truth of the (notoriously open) Evasiveness Conjecture. Does this constructively imply the negation of the Pizzazz-conjecture? Remarks. The relevant statements are, by themselves, so natural, and transparently explained in the links given, that repeating them here seems ...