On Mordell–Weil groups of isotrivial abelian varieties over function fields

Abstract

We show that the Mordell–Weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant, provided the discriminant has singularities in CM class introduced here. This class of singularities includes all unibranched plane curves singularities. As a corollary, we describe a family of simple Jacobians over the field of rational functions in two variables for which the Mordell–Weil rank is arbitrarily large.

Appendix: Jacobians of Belyi covers

In this appendix we shall prove the Lemma 3.14 i.e. that the Jacobians of Belyi cyclic covers are abelian varieties of the CM type. Though the Lemma 3.14 is apparently not new (cf. [14, 19]) the proof given below for convenience contains explicit formulas for the eigenvalues of the automorphisms of Belyi covers acting on the space of holomorphic 1-forms.

Proof

(of Lemma 3.14) We claim that a generator of the group of deck transformations of a cyclic Belyi cover acting on $H_1$ does not have multiple eigenvalues. Once this is established, an argument as in the proof of the Theorem 3.7, shows that for the Jacobian of such cover one has $2dim J=dim End^{\circ }(J)$ i.e. $J$ has a CM type.

Let $C \rightarrow \mathbb{P }^1$ be a Belyi cyclic cover and let $d$ be its degree, i.e. the group of roots of unity of degree $\mu _d$ acts on $C$ with three points having non-trivial stabilizers. Let $a,b,c$ be the indices these stabilizers in $\mu _d$. As a model of such Belyi cover (suitable for our calculations below, cf. Proof of Lemma 6.1), one can choose the normalization of plane curve:

$$\begin{aligned} y^d=x^a(x-z)^bz^c , \quad a+b+c=d. \end{aligned}$$

(46)

The action of $\mu _d$ is given by $T: (x,y,z) \rightarrow (x,e^{2 \pi i \over d}y,z)$. Let $T_*$ be the induced map on $H_1(X,\mathbb{C })$. Now, the proper preimage for the map

$$\begin{aligned} (x,y) \rightarrow (x^d,y^bx^a) \end{aligned}$$

(47)

of the affine model of (46), i.e.

$$\begin{aligned} y^d=x^a(x-1)^b \end{aligned}$$

(48)

yields a curve which has a component the Fermat curve

$$\begin{aligned} y^d=x^d-1. \end{aligned}$$

(49)

Since the Jacobian of Fermat curve is a product of abelian varieties of CM type (cf. [19]) this implies that the same is the case for cyclic Belyi covers. $\square$

The following allows one effectively to calculate the CM type of local Albanese varieties in many cases. These formulas extend the special case presented in [43]. We have the following:

Lemma 6.1

• 1. The multiplicity of the eigenvalue $\omega _d^j=e^{2 \pi \sqrt{-1} j\over d}$ of $T_*$ acting on the space of holomorphic 1-forms of Belyi cyclic cover as above is equal to:

  $$\begin{aligned} -\left(\left[-{{aj}\over d}\right]+\left[-{{bj}\over d}\right]+\left[{{(a+b)j} \over d}\right]+1\right) \end{aligned}$$

  (50)

  where $[\cdot ]$ denotes the integer part. In particular this multiplicity is equal either to zero or one.

• 2. Let $gcd(a,b,c,d)=1$ (i.e. the Belyi cover is irreducible). Then the characteristic polynomial of the deck transformation acting on $H_1$ is given by

  $$\begin{aligned} \Delta (t)={{(t^{d}-1)(t-1)^2} \over {{(t^{gcd(a,d)}-1)} {(t^{gcd(b,d)}-1)}{(t^{gcd(c,d)}-1)}}} \end{aligned}$$

  (51)

Proof

(of Lemma 6.1) First note that the indices of stabilizers for the branching points of the cover (46) as the subgroups of the covering group are $gcd(a,d),gcd(b,d), gcd(c,d)$ respectively. Hence Riemann–Hurwitz formula yields that the genus of $C$ is given by (cf. [17])

$$\begin{aligned} g={{d-gcd(a,d)-gcd(b,d)-gcd(c,d)+2} \over 2}. \end{aligned}$$

(52)

We shall represent explicitly the cohomology classes of $H^0(\Omega ^1_C)$ and calculate the action of covering group on holomorphic 1-forms. Recall that the space of holomorphic 1-forms on a plane curve of degree $d$ can be identified with the space of adjoint curves of degree $d-3$, i.e. the curves of degree $d-3$ which equations at each singular point satisfy the adjunction conditions or equivalently belong to the adjoint ideal of this singularity. This can be made explicit since any holomorphic 1-form can be written as the residue of 2-form on its complement, i.e. as

$$\begin{aligned} {{ P(x,y)dx} \over {y^{d-1}}} \quad \mathrm{deg}P \le d-3 \end{aligned}$$

(53)

The curve (46) may have singular points only at $(0,1),(1,0),(1,1)$ and near each the local equation is equivalent to $x^l+y^d=0$ (by abuse of language we shall refer to these points as “singular” even if the curve is smooth there). To calculate the number of adjunction conditions we shall use the following (cf. [29]):

Proposition 6.2

The conditions of adjunction for the singularity $y^d+x^l$ are the vanishing of the coefficients of monomials $x^ij^j$ such that $(i+1,j+1)$ is below or on the diagonal of the rectangle with vertices $(0,0),(0,d),(l,0),(l,d)$. The number of adjunction conditions for singularity $y^d+x^l$ is equal to

$$\begin{aligned} {{(d-1)(l-1)+gcd(d,l)-1} \over 2} \end{aligned}$$

(54)

This implies that the dimension of the space of curves of degree $d-3$ satisfying the conditions of adjunction at all three singularities is greater or equal:

$$\begin{aligned}&{{(d-1)(d-2)} \over 2} -{{(d-1)(a-1)+gcd(d,a)-1} \over 2}\\&\qquad - {{(d-1)(b-1)+gcd(d,b)-1} \over 2} -{{(d-1)(c-1)+gcd(d,c)-1} \over 2}\\&\quad = {{d+2-gcd(a,d)-gcd(b,d)-gcd(c,d)}\over 2}. \end{aligned}$$

Comparison of this with the genus formula (52) shows that the conditions of adjunction imposed by three singular points are independent, i.e. one has the exact sequence

$$\begin{aligned} 0 \rightarrow H^0(\Omega ^1_{\tilde{C}}) \rightarrow H^0(\mathbb{P }^2,\Omega ^2_{\mathbb{P }^2}(d)) \rightarrow \oplus _{s \in Sing C} M_s \rightarrow 0 \end{aligned}$$

where $\tilde{C}$ is the normalization of $C$ and $M_s$ is the quotient of the local ring of singular point by the adjoint ideal.

To calculate the action of $T_*$ on $H^1(\tilde{C},\mathbb{C })$, we shall use the identification (53) of adjoints with the forms and that the action of $T^*$ on the monomials is given by $g(x^iy^j)=\omega _d^jx^iy^j$. Also note that the cardinality of the set of solutions to linear inequality (for a fixed $j$) is given as follows:

$$\begin{aligned} Card \{ i \vert 0 < i, \ di+aj \le da\}=a+\left[-{{aj}\over d}\right] \end{aligned}$$

(55)

The multiplicity of the eigenvalue corresponding to the monomial $x^iy^{j-1}$ (i.e. $\omega _d^{j-1}$) in representation of $\mu _d$ in $H^0(\mathbb{P }^2,\Omega ^2_{\mathbb{P }^2}(d))=H^0(\Omega ^1_{\bar{C}})$ where $\bar{C}$ is a smoothing of $C$ is $\mathrm{Card} \{ i \vert 0 < i, i+j-1 \le d-3\} =d-1-j$. Hence the multiplicity of the eigenvalue $\omega ^j$ is equal to

$$\begin{aligned}&d-1-j-a-\left[-{{aj}\over d}\right] -b-\left[-{{bj}\over d}\right]-c-\left[-{{cj}\over d}\right]=\nonumber \\&\quad = -\left(\left[-{{aj}\over d}\right]+\left[-{{bj}\over d}\right]+\left[{{(a+b)j} \over d}\right]+1\right). \end{aligned}$$

(56)

The last assertion of Lemma 6.1, i.e. that the multiplicity does not exceed 1, follows from the property $[x+y] \le [x]+[y]+1$.

The formula for the characteristic polynomial can be derived using the additivity of zeta function similarly to the expression for the euler characteristic obtained earlier. $\square$

To finish a proof of Lemma 3.14, just note that absence of multiple eigenvalues implies that the Jacobian must have a CM type (as in the Proof of Theorem 3.7).