On the arithmetic of a family of superelliptic curves

Abstract

Let p be a prime, let r and q be powers of p, and let a and b be relatively prime integers not divisible by p. Let $C/{\mathbb {F}}_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$, and let J be the Jacobian of C. By work of Pries and Ulmer (Trans Am Math Soc 368(12):8553–8595, 2016), J satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon and Ulmer (Pacific J Math 305(2):597–640, 2020) , we compute the L-function of J in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of J appearing in BSD, including the rank of the Mordell–Weil group $J({\mathbb {F}}_{r}(t))$, the Faltings height of J, and the Tamagawa numbers of J in terms of the parameters a, b, q. For any p and r, we show that for certain a and b depending only on p and r, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as q varies through powers of p. Under a different set of criteria on a and b, we prove that the order of the Tate–Shafarevich group grows exponentially fast in q as $q \rightarrow \infty$.

Conductor computations

Recall that $N_{J} \in \text {Div}(\mathbb {P}^1)$ is the conductor divisor of J/K.

Proposition A.1

We prove the statement from Theorem 4.1 regarding the global degree b(J) of the L-function L(J, T):

$$\begin{aligned} b(J) = \deg (N_J) - 4g. \end{aligned}$$

Proof

We begin by defining the conductor divisor $N_J$ as a divisor on the base $\mathbb {P}^1$. The action of inertia $I_v$ on the $\ell$-adic Tate module $V_\ell$ is tame.¹ For any place v of K, define

$$\begin{aligned} f(v):=\dim (V_\ell ) - \dim (V_\ell ^{I_v}), \end{aligned}$$

and let the conductor of J be the divisor $N_J:=\sum _{v}f(v)v$ on $\mathbb {P}^1$. By [25], $f(v)=0$ whenever v is a place of good reduction for J. Plugging in $\dim (V_\ell )=2g$ gives

$$\begin{aligned} \deg (N_J)= \sum _{v \text { bad reduction}}(2g - \dim (V_\ell ^{I_v})) \deg v, \end{aligned}$$

where the sum is over places v of K where J has bad reduction. Now, we investigate the L-function and see how its global degree relates to $\deg N_J$. Begin with the definition:

$$\begin{aligned} L(J,T):= \prod _{v}\det (1 - T\textrm{Fr}_v^{-1}|V_\ell ^{I_v})^{-1}. \end{aligned}$$

This product can be split up into products over good and bad places of C:

$$\begin{aligned} L(J,T):= \prod _{\text {good }v}\det (1 - T\text {Fr}_v^{-1}|V_\ell ^{I_v})^{-1}\prod _{\text {bad }v}\det (1 - T\text {Fr}_v^{-1}|V_\ell ^{I_v})^{-1}. \end{aligned}$$

Let ${\tilde{L}}(J,T):= \underset{\text {good }v}{\prod }\det (1 - T\text {Fr}_v^{-1}|V_\ell ^{I_v})^{-1}$. This gives a decomposition of the global degree:

$$\begin{aligned} \deg (L(J,T)) = \deg ({\tilde{L}}(J,T)) - \underset{\text {bad }v}{\sum }\dim (V_\ell ^{I_v}). \end{aligned}$$

Since L(J, T) is rational, and since the sum $\underset{\text {bad }v}{\sum }\dim (V_\ell ^{I_v})$ is finite, the “complement” ${\tilde{L}}(J,T)$ is also rational. From here, we need a more precise formula for $\deg ({\tilde{L}}(J,T))$. Let U denote the affine open subset of $\mathbb {P}^1$ above which J has good reduction. Since U is a punctured $\mathbb {P}^1$, by the étale-singular cohomology comparison theorem, we have

$$\chi (U,\overline{\mathbb {Q}_\ell }):=\dim H^0(U,\overline{\mathbb {Q}}_\ell )-\dim H^1(U,\overline{\mathbb {Q}}_\ell )+\dim H^2(U,\overline{\mathbb {Q}}_\ell )=2-2g(\mathbb {P}^1)-r,$$

where $g(\mathbb {P}^1)$ is the genus of $\mathbb {P}^1$ and r is the number of geometric points over which J has bad reduction. That is, r is the sum of the degrees of places of bad reduction for J, namely $r=\underset{\text {bad }v}{\sum } \deg v$. Therefore $\chi (U,\overline{\mathbb {Q}_\ell })=2-r$.

The Grothendieck–Ogg–Shafarevich formula (see [4]) yields that

$$\begin{aligned} \chi (U,\mathcal {F})=\chi (U,\overline{\mathbb {Q}_\ell })\cdot {{\,\textrm{rank}\,}}(\mathcal {F})-\sum _{x\in \mathbb {P}^1\setminus U}({{\,\textrm{rank}\,}}(\mathcal {F})+Sw_x(\mathcal {F})), \end{aligned}$$

where in our case $\mathcal {F}=V_\ell$, which is a lisse $\ell$-adic sheaf of rank $\dim V_\ell = 2g$ on U. Since the action of inertia on $V_\ell$ is tame (see [28, Corollary 2, p. 497]), this implies that

$$\begin{aligned} \chi (U,\mathcal {F})=\chi (U,\overline{\mathbb {Q}_\ell })\cdot {{\,\textrm{rank}\,}}(\mathcal {F})=2g(2-r). \end{aligned}$$

Now, since $\deg {\tilde{L}}(J,T)=-\chi (U,\mathcal {F})$, we deduce that $\det {\tilde{L}}(J,T) =-2g(2-r)$. Putting this back into the equation for $\deg L(J,T)$ gives

$$\begin{aligned} \deg (L(J,T))&= \deg ({\tilde{L}}(J,T)) - \sum _{\text {bad } v}\dim (V_\ell ^{I_v}) =-4g + \sum _{\text {bad } v}2g - \sum _{\text {bad } v}\dim (V_\ell ^{I_v})\\&= \sum _{\text {bad } v}(2g -\dim (V_\ell ^{I_v})) - 4g = \deg (N_J)-4g. \end{aligned}$$

$\square$