4 problems

For integers $A,B$, let $E_{A,B}$ be the elliptic curve … with no prime $p$ satisfying both $p^4\mid A$ and $p^6\mid B$. Define its naive height by … and write $N(E)$ for its condu…

Let $(\mathcal N,\mathcal Q)$ be an arithmetically compatible sequence of pairs $(N,Q)$ of Type I. Let $\mathcal E$ be the set of rational newforms lying in $S_2(N)$ for some…

Fix $k,r$ and $0\le m<r$, and fix primes $p_1<\dots<p_m$. Let $\mathcal N$ be the set of squarefree levels $N=p_1\dots p_r$ with $r$ prime factors, where the first $m$ primes are f…

Let $(\mathcal N,\mathcal Q)$ be an arithmetically compatible sequence of $(N,Q)$'s of Type I, II or III as above, and fix a weight $k$. Let $\mathcal F$ be the family of newforms…