Frobenius in $W(E_8)$ for $y^2=x^3+x+D(t)$ over ${\mathbb{F}}_q(t)$: ranks, Tate--Shafarevich groups and trace statistics

Abstract

Let $q$ be a power of a prime $p\ge 5$ and let ${\mathcal{F}}_5(q)$ be the set of curves $E_D:y^2=x^3+x+D(t)$ over ${\mathbb{F}}_q(t)$ with $D$ monic of degree $5$ and $4+27D^2$ squarefree. Each $E_D$ is the generic fibre of a rational elliptic surface with Mordell--Weil lattice $E_8$, Frobenius acts on it by some $w_D\in W(E_8)$, and $L(E_D,T)=\det (1-q{w}_{D}T)$ (Shioda, Oguiso--Shioda, Ulmer). From known results (Galois descent, Grothendieck's $\mathrm{Sha}\cong \mathrm{Br}$, Artin, Milne, Urabe) one reads off $E_D({\mathbb{F}}_q(t))=E_8^{w_D}$, $\mathrm{Sha}(E_D)\cong (E_8/(1-w_D)E_8{)}_{\mathrm{tors}}$ and $|\mathrm{Sha}|\cdot \mathrm{Reg}=\det {}^{\prime }(1-w_D)$; by the class tables of Urabe and Karras, $|\mathrm{Sha}|$ is not determined by $L(E_D,T)$ at positive rank. The main new result is that the monodromy of this $4$-parameter family is $W(E_8)$ in every characteristic $p\ge 5$ (computer-assisted). With Deligne's theorem, and Katz's Betti bounds for uniformity in $p$, $w_D$ is equidistributed in $W(E_8)$ with error $O(q^{-1/2})$ as $q\to \infty$ through all prime powers; e.g. the mean degree-$1$ trace over the rank-$r$ curves, which equals $-\mathbb{E}[\mathrm{tr}{w}_D\mid {\mathrm{mult}}_1(w_D)=r]$, tends to $8/29$ at rank $0$. The constant is astronomical. We prove first-order ($1/q$) formulas for the moments of $\mathrm{tr}{w}_D$ of order $\le 3$; their coefficients depend on $q$ through ${\chi }_q(-3)$ and, at order $3$, through a Frobenius trace on a $2$-dimensional $H^1$. We conjecture a first-order law for the whole class distribution; one of its four terms involves the CM newform $\eta (4z{)}^6$ (checked for all primes below $3000$). The law fits exact class counts at all prime powers $q\le 41$, including out-of-sample predictions at $q=37,41$, some of which (rank statistics at $q=41$) missed.

arXiv commentsubstantially revised; title changed (v1: "Murmurations of Elliptic Curves over Function Fields"). The computed data of v1 were incorrect; all corrections are listed in the ancillary file CORRECTIONS.md. 34 pages