Seshadri constants on principally polarized abelian surfaces with real multiplication

Abstract

Seshadri constants on abelian surfaces are fully understood in the case of Picard number one. Little is known so far for simple abelian surfaces of higher Picard number. In this paper we investigate principally polarized abelian surfaces with real multiplication. They are of Picard number two and might be considered the next natural case to be studied. The challenge is to not only determine the Seshadri constants of individual line bundles, but to understand the whole Seshadri function on these surfaces. Our results show on the one hand that this function is surprisingly complex: on surfaces with real multiplication in $\mathbb {Z}[\sqrt{e}]$ it consists of linear segments that are never adjacent to each other—it behaves like the Cantor function. On the other hand, we prove that the Seshadri function is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.

1 Introduction

The purpose of this paper is to contribute to the study of Seshadri constants on abelian surfaces. Recall that for an ample line bundle L on a smooth projective variety X, the Seshadri constant of L at a point $x\in X$ is by definition the real number

On abelian varieties, where this invariant is independent of the chosen point x, we write simply $\varepsilon (L)$. Seshadri constants are highly interesting invariants for numerous reasons: They are related to minimal period lengths [1, 14], to syzygies [13, 16], and they govern quite generally the geometry of linear series in many respects [11, 12] (we refer to [15, Chapt. 5] and [6] for more background on Seshadri constants).

On abelian surfaces, Seshadri constants are fully understood in the case of Picard number $\rho =1$ [5]. For $\rho >1$, only self-products of elliptic curves have been studied, while the important case of simple abelian surfaces is completely unexplored so far. In contrast to the case of $\rho =1$, the challenge on these surface is not only to determine the Seshadri constant of one ample line bundle, but to understand the behavior of the Seshadri function,

$$\begin{aligned} \varepsilon :\mathrm{Amp}(X)\rightarrow \mathbb {R}, \quad L\mapsto \varepsilon (L), \end{aligned}$$

which associates to each ample line bundle its Seshadri constant. To our knowledge, there are—also beyond abelian surfaces—hardly any cases where this function is known explicitly, the exception being certain self-products $E\times E$ of elliptic curves [4]. In general, the Seshadri function of an abelian variety is known to be concave and continuous [4, Prop. 3.1], but at present it is unclear what kind of behavior to expect beyond these basic properties.

We attack this problem on abelian surfaces of Picard number $\rho =2$, which seems to be the natural next case to investigate. As the Seshadri function is homogeneous, it is completely determined by its values on a cross-section of $\mathrm{Amp}(X)$. So, when $\mathrm{Amp}(X)$ is two-dimensional, we may consider it as a function $\varepsilon :I\rightarrow \mathbb {R}$ on an interval $I\subset \mathbb {R}$. We always take this point of view when we speak of the Seshadri function.

For clarity of exposition let us introduce a piece of terminology:

Definition

Let $I\subset \mathbb {R}$ be an interval. A function $f:I\rightarrow \mathbb {R}$ is called broken linear, if it is continuous and there is a non empty and nowhere dense subset $M\subset I$ such that the following holds:

• (i) Around every point of $I{\setminus } M$ there is an open interval, contained in $I{\setminus } M$, on which f is linear.

• (ii) If $I_1$ and $I_2$ are maximal open subintervals of I on which f is linear, then $I_1$ and $I_2$ are contained in $I{\setminus } M$, and $I_1$ and $I_2$ are not adjacent to each other (i.e., an endpoint of $I_1$ is never an endpoint of $I_2$).

Note that these conditions imply that M is a perfect set (i.e., that every point of M is an accumulation point of M) and, thus, M is uncountable. More concretely, condition (ii) implies that whenever a linear piece of f ends (i.e., one of the maximal subintervals mentioned in the definition), then no other linear piece begins at that point, but instead there is a sequence of linear pieces converging to that point. And the same applies to the converging pieces: each of them is again approached by a sequence of pieces. The Cantor function (see e.g. [10]) is an example of a broken linear function (in which case the Cantor set is the perfect set M).

Our first result shows that on abelian surfaces with real multiplication, the Seshadri function is of the same baffling complexity as the Cantor function:

Theorem A

Let X be a principally abelian surface, whose endomorphism ring is isomorphic to $\mathbb {Z}[\sqrt{e}]$ for some non-square integer $e>0$. Then the Seshadri function of X is broken linear.

This result is in stark contrast to what had been observed so far: When E is a general elliptic curve, then the restriction of the Seshadri function on $E\times E$ to any rational line is a piecewise linear function, in the usual sense that each piece is adjacent to another piece [4]. The situation in Theorem A is at the other extreme: At no point are two pieces connected to each other.

The endomorphism ring of an abelian surface with real multiplication is an order of a quadratic number field $\mathbb {Q}(\sqrt{d})$. As the integer e appearing in Theorem A is not required to be square-free, the only orders not covered there are those of the form $\mathbb {Z}[\frac{1}{2}+\frac{1}{2}\sqrt{e}]$, where $e\equiv 1 \;(\mathrm{mod}~4)$. Surfaces with these endomorphism rings add another level of complexity: We show that on certain surfaces of this type, every line bundle has only one submaximal curve (which then computes its Seshadri constant), while there also exist surfaces of this type carrying line bundles with two submaximal curves (see Propositions 5.2 and 5.4). Interestingly, the conclusion of Theorem A extends to the former surfaces (see Theorem 3.7), whereas on the latter surfaces there exist boundary points of linear segments which are accumulation points, as well as boundary points where linear segments meet (see Remark 3.8).

The discussion so far has shown how complex and subtle the Seshadri function on surfaces with real multiplication is. Our next result states that globally it has more structure than one might expect at this point:

Theorem B

There exists a decomposition of the ample cone into infinitely many subcones $\mathcal {C}_k$, $k\in \mathbb {Z}$, such that the group G of isometries of $\mathrm{NS}(X)$ that leave the Seshadri function on $\mathrm{Amp}(X)$ invariant acts transitively on the set of subcones. In particular, the values of the Seshadri function on any subcone of the subcones $\mathcal {C}_k$ completely determine the Seshadri function on the entire ample cone.

There are only few known cases where one has effective computational access to the Seshadri constants of all line bundles on the surface (the self-product $E\times E$ of a general elliptic curve being an exception again). Our methods provide such computational access for the surfaces studied here.

Theorem C

There is an algorithm that computes the Seshadri constant of every given ample line bundle on principally polarized abelian surfaces with real multiplication.

The algorithm enables us to efficiently compute Seshadri functions and thus to provide a graphical representation for any given endomorphism ring (see examples in Sect. 4). Also, the analysis of the method underlying the proof of Theorem C allows us to answer the question as to which data in fact determine the Seshadri function. A priori, the function could depend on the individual surface (or rather, on its isomorphism class). However, the numerical data entering the computation ultimately stems from the endomorphism ring, and this implies:

Corollary D

Let X and Y be principally polarized abelian surfaces with real multiplication, such that $\mathrm{End}(X)\simeq \mathrm{End}(Y)$. Then, in suitable linear coordinates on $\mathrm{NS}(X)$ and $\mathrm{NS}(Y)$, their Seshadri functions coincide.

Note that the assumption that $\mathrm{End}(X)$ and $\mathrm{End}(Y)$ be isomorphic is strictly weaker than requiring that X and Y be isomorphic. In fact, the corollary shows that only countably many Seshadri functions occur, while the surfaces vary in two-dimensional families.

Concerning the organization of this paper, we start in Sect. 1 by establishing crucial properties of Pell divisors and submaximal curves. We study in Sect. 2 the intervals on which curves can be submaximal. Sect. 3 is devoted to the proofs of Theorems A and C, as well as Corollary D. In Sect. 4 we study the decomposition of the ample cone and prove Theorem B. Finally, in Sect. 5 we investigate in terms of $\mathrm{End}(X)$ the question on which surfaces there are line bundles with two submaximal curves.

We would like to thank Robert Lazarsfeld for valuable suggestions concerning the exposition.

Throughout we work over the field of complex numbers.

2 Pell divisors and submaximal curves on abelian surfaces

As in the introduction, we refer to [15, Chapt. 5] and [6] for background on Seshadri constants. Let us just fix a few matters of terminology here. When we speak of the general upper bound, then we mean the bound $\varepsilon (L,x)\leqslant \sqrt{L^2}$, which is valid for every ample line bundle L on a smooth projective surface S and for every point $x\in S$. An effective divisor D on S is called submaximal (for L at x), if $L\cdot D/\mathrm{mult}_x D<\sqrt{L^2}$. If an irreducible curve $C\subset S$ satisfies the equation $L\cdot C/\mathrm{mult}_x C=\varepsilon (L,x)$, then we say that C computes $\varepsilon (L,x)$. An irreducible curve which computes $\varepsilon (L,x)$ for some ample line bundle L on S will be called a Seshadri curve on S.

It was shown in [5] that on an abelian surface of Picard number one, with ample generator L of the Néron–Severi group, there is for suitable $k\geqslant 1$ a divisor $D\in |2kL|$ that computes the Seshadri constant of L. The number k and the multiplicity of D at 0 are governed by a Pell equation. We will see that a suitable notion of Pell divisors (in the sense of the subsequent definition) also play a crucial role in the present investigation. The results in this section work on all abelian surfaces and do not require that the surface has real multiplication.

Definition 1.1

Let A be an abelian surface, and let L be an ample primitive symmetric line bundle such that $\sqrt{L^2}\notin \mathbb {Z}$. Consider the Pell equation

$$\begin{aligned} \ell ^2 - L^2\cdot k^2 = 1 \end{aligned}$$

and let $(\ell ,k)$ be its primitive solution. A divisor $D\in |2kL|^+$ with $\mathrm{mult}_0 D\geqslant 2\ell$ is called a Pell divisor for L.

Here $|2kL|^+$ denotes the linear subsystem of even divisors in |2kL|, i.e, those defined by even theta functions (see [7, Sect. 4.7]). It will be convenient to extend the notion of Pell divisors to non-primitive bundles, and even to $\mathbb {Q}$-divisors:

Definition 1.2

Let A be an abelian surface, and let M be any ample $\mathbb {Q}$-line bundle on A such that $\sqrt{M^2}\notin \mathbb {Q}$. Write $M=q L$ with a primitive ample line bundle L and $q\in \mathbb {Q}$. A Pell divisor for M is then by definition a Pell divisor for L.

It was shown in [2, Theorem A.1] that Pell divisors exist for every ample line bundle L with $\sqrt{L^2}\notin \mathbb {Z}$. Their crucial feature is that they are submaximal for L. By contrast, the existence of submaximal divisors is not guaranteed when $\sqrt{L^2}\in \mathbb {Z}$. However, a dimension count shows that for such bundles there exist divisors $D\in |2L|^+$ satisfying in any event the weak inequality $L\cdot D / \mathrm{mult}_0D\leqslant \sqrt{L^2}$.

We will see that on abelian surfaces with real multiplication it is almost never true that $\varepsilon (L)$ is computed by a Pell divisor of L. However, it will turn out that $\varepsilon (L)$ is always computed by a Pell divisor of some ample bundle on L, whenever $\sqrt{L^2}$ is irrational. It is for this reason that Pell divisors are crucial players in the present investigation.

The following statement will prove to be a valuable tool, as it provides strong restrictions on submaximal curves. Also, it exhibits a situation where Pell divisors are unique.

Proposition 1.3

Let A be an abelian surface, and let $C\subset A$ be an irreducible curve that is submaximal for some ample line bundle. Then, putting $m=\mathrm{mult}_0C$, one has

$$\begin{aligned} C^2-m^2=-1 \quad \text{ or }\quad C^2-m^2=-4. \end{aligned}$$

Furthermore, suppose that C is not an elliptic curve, and write $\mathcal {O}_A(C)=pM$ with a primitive ample bundle M and an integer $p>0$. Then $\sqrt{M^2}$ is irrational, and letting $(\ell _0,k_0)$ be the primitive solution of the Pell equation $\ell ^2-M^2 k^2=1$, we have:

• (i) If $C^2-m^2=-1$, then the divisor 2C is the only Pell divisor for M and $(\ell _0,k_0)=(m,p)$.

• (ii) If $C^2-m^2=-4$, then the curve C is the only Pell divisor for M and $(2\ell _0,2k_0)=(m,p)$. In this case, the origin is the only halfperiod that lies on C.

Proof

The first half of the following argument is implicit in the proof of [3, Thm. 2]. To provide easier access, we briefly make it explicit here. As the claim on $C^2-m^2$ is certainly true for elliptic curves, we may assume that C is non-elliptic, and hence that $\mathcal {O}_A(C)$ is ample. The assumption that C is submaximal for some ample line bundle L then implies that C is submaximal also for $\mathcal {O}_A(C)$, and in fact computes $\varepsilon (\mathcal {O}_A(C))$ (see [4, Prop. 1.2]). Therefore C must be symmetric and therefore descends to a $(-2)$-curve on the smooth Kummer surface of A (cf. proof of [5, Thm. 6.1]). The multiplicities $m_i=\mathrm{mult}_{e_i}(C)$ at the sixteen halfperiods $e_i$ of A therefore satisfy the equation

$$\begin{aligned} C^2 - \sum _{i=1}^{16} m_i^2 = -4 . \end{aligned}$$

(1)

Putting $m=m_1$, one shows as in the proof of [3, Thm. 1.2] that only the two cases

$$\begin{aligned} C^2 - m^2 = -1 \quad \text{ or }\quad C^2 - m^2 = -4, \end{aligned}$$

(2)

are possible. This proves the first statement in the proposition.

Write now $\mathcal {O}_A(C)=pM$ with a primitive ample bundle M and $p>0$. It follows from Eq. (2) that $C^2$ cannot be a perfect square, and hence that $\sqrt{M^2}$ is irrational. In the first case of (2), the pair (m, p) satisfies the Pell equation $m^2-M^2p^2=1$. The minimality of the solution $(\ell _0,k_0)$ implies then that $m\geqslant \ell _0$ and $p\geqslant k_0$. On the other hand, as C computes $\varepsilon (M)$, we have for every Pell divisor $P\in |2k_0M|$ of L,

$$\begin{aligned} \frac{M\cdot C}{m} \leqslant \frac{M\cdot P}{\mathrm{mult}_0 P}\leqslant \frac{M\cdot P}{2\ell _0} \end{aligned}$$

This implies $\frac{p}{m} \leqslant \frac{k_0}{\ell _0}$. Using the fact that both pairs (m, p) and $(\ell _0,k_0)$ solve the Pell equation, we find $m\leqslant \ell _0$, and hence $(m,p)=(\ell _0,k_0)$. So we have $P=2C$ in this case.

In the second case of (2), the number m is clearly even. But also p is even in this case, because all multiplicities $m_i$ are even (since we have $(m_1,\ldots ,m_{16})=(m,0,\ldots ,0)$). Therefore $\mathcal {O}(C)$ is totally symmetric, and it can therefore be written as an even multiple of another bundle (see [7, Sect. 2, Cor. 4]). The upshot of this argument is that the pair $(\frac{m}{2},\frac{p}{2})$ satisfies the Pell equation $(\frac{m}{2})^2-M^2(\frac{p}{2})^2=1$. The minimality assumption implies then that $\frac{m}{2}\geqslant \ell _0$ and $\frac{p}{2}\geqslant k_0$. But C must be a component of any Pell divisor $P\in |2k_0M|$ by [5, Lemma 6.2], and so $p=2k_0$ and $m=2\ell _0$. So we have $P=C$ in this case. $\square$

Further, we show how two curves can intersect if they are submaximal for the same bundle:

Proposition 1.4

Let A be an abelian surface, and let $C_1$ and $C_2$ be two irreducible curves on A that are submaximal for the same ample line bundle L on A, i.e.,

$$\begin{aligned} \frac{L\cdot C_i}{\mathrm{mult}_{0}(C_i)}<\sqrt{L^2} \end{aligned}$$

for $i=1,2$. Then, putting $m_i=\mathrm{mult}_0(C_i)$, we have

$$\begin{aligned} C_1\cdot C_2 = m_1 m_2, \end{aligned}$$

i.e., the curves $C_1$ and $C_2$ meet only at the origin, and their tangent cones have no common components there.

Proof

Consider the blow-up $f:Y=\mathrm{Bl}_0(A)\rightarrow A$, let E be its exceptional divisor, and let $C_i'\subset Y$ be the proper transform of $C_i$. For rational numbers $t<\sqrt{L^2}$, the $\mathbb {Q}$-divisor

$$\begin{aligned} B := f^*L-tE \end{aligned}$$

is big, because $B^2=L^2-t^2>0$ and $B\cdot f^*L=L^2>0$. If we take t strictly between $\max \left\{ \frac{L\cdot C_1}{m_1},\frac{L\cdot C_2}{m_2}\right\}$ and $\sqrt{L^2}$, then we moreover have

$$\begin{aligned} B\cdot C_i' = (f^*L-tE)\cdot (f^*C_i-m_iE) =L\cdot C_i - tm_i < 0. \end{aligned}$$

As a consequence, both $C_1'$ and $C_2'$ must be contained in the negative part of the Zariski decomposition of B, and hence their intersection matrix is negative definite. This implies that $C_1'^2 C_2'^2 > (C_1'C_2')^2$, i.e.,

$$\begin{aligned} (C_1^2-m_1^2)(C_2^2-m_2^2) > (C_1\cdot C_2 - m_1m_2)^2. \end{aligned}$$

(3)

We know from Proposition 1.3 that $C_i^2-m_i^2\in \left\{ -1,-4\right\}$. Let us first consider the case $C_1^2-m_1^2=C_2^2-m_2^2=-1$. Then inequality (3) directly implies $C_1\cdot C_2-m_1 m_2=0$. Suppose next that $C_1^2-m_1^2=-4$ and $C_2^2-m_2^2=-1$. In that case inequality (3) tells us that

$$\begin{aligned} C_1\cdot C_2 - m_1m_2 < 2 , \end{aligned}$$

since we have in any event $C_1\cdot C_2 - m_1m_2 \geqslant 0$ because of the intersection inequality. Using now Proposition 1.3, we see that $m_1$ is an even number and that $C_1\equiv 2B_1$ for some line bundle $B_1$ on A. So we obtain $B_1\cdot C_2 - \frac{m_1}{2} m_2 < 1$ and hence $B_1\cdot C_2=\frac{m_1}{2} m_2$, which implies $C_1\cdot C_2 = m_1m_2 = 0$, as claimed. Finally, if both $C_1^2-m_1^2$ and $C_2^2-m_2^2$ equal $-4$, then we get

$$\begin{aligned} C_1\cdot C_2 - m_1m_2 < 4 \end{aligned}$$

from inequality (3), and we have $C_1\equiv 2B_1$, $C_2\equiv 2B_2$. As both $m_1$ and $m_2$ are even, this yields $B_1\cdot B_2 - \frac{m_1}{2} \frac{m_2}{2} = 0$, and this implies the assertion. $\square$

3 Submaximal curves on intervals

Abelian surfaces with real multiplication. Let X be a simple abelian surface with real multiplication, i.e., such that $\mathrm{End}_\mathbb {Q}(X)=\mathbb {Q}(\sqrt{d})$ for some square-free integer $d\geqslant 2$. The endomorphism ring is an order in $\mathrm{End}_\mathbb {Q}(X)$ and hence of the form $\mathrm{End}(X)=\mathbb {Z}+f\omega \mathbb {Z}$, where $f\geqslant 1$ is an integer and

$$\begin{aligned} \omega ={\left\{ \begin{array}{ll} \sqrt{d} &{} \text{ if } d\equiv 2,3 \;(\mathrm{mod}~4) \\ {\textstyle \frac{1}{2}}(1+\sqrt{d}) &{} \text{ if } d\equiv 1 \;(\mathrm{mod}~4). \end{array}\right. } \end{aligned}$$

For our purposes an alternative distinction of the possible cases will be more convenient:

• Case 1: $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$, with a non-square integer $e>0$.

• Case 2: $\mathrm{End}(X)=\mathbb {Z}[\frac{1}{2}+\frac{1}{2}\sqrt{e}]$ with a non-square integer $e>0$ such that $e\equiv 1\;(\mathrm{mod}~4)$.

If X carries a principal polarization $L_0$, then we have an isomorphism $\varphi :\mathrm{NS}(X)\rightarrow \mathrm{End}^\mathrm{sym}(X)=\mathrm{End}(X)$. It provides us with a lattice basis of $\mathrm{NS}(X)$, given by $L_0=\varphi ^{\smash -1}(1)$ and $L_{\infty }:=\varphi ^{\smash -1}(\sqrt{e})$ (resp. $\varphi ^{\smash -1}({\textstyle \frac{1}{2}}+{\textstyle \frac{1}{2}}\sqrt{e})$). The intersection matrix of this basis is

(4)

This follows by considering the characteristic polynomials of $\sqrt{e}$ and ${\textstyle \frac{1}{2}}+{\textstyle \frac{1}{2}}\sqrt{e}$ in $\mathbb {Q}(\sqrt{e})$ (which coincides with the analytic characteristic polynomial of the endomorphism) and applying [7, Prop. 5.2.3].

Using the Nakai–Moishezon criterion (in the version of [7, Cor. 4.3.3]), and the fact that X does not contain any elliptic curves, we find:

Lemma 2.1

Let L be a line bundle on X with numerical class given by $L=aL_0+bL_\infty$ for $a,b\in \mathbb {Z}$. If $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$, then L is ample if and only if

$$\begin{aligned} a>0 \quad \text{ and }\quad a^2-eb^2>0, \end{aligned}$$

and if $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}}+{\textstyle \frac{1}{2}}\sqrt{e}]$, then L is ample if and only if

$$\begin{aligned} a>0 \quad \text{ and }\quad a^2+ab+{\textstyle \frac{1-e}{4}}\,b^2>0. \end{aligned}$$

In either case, L is ample if and only if $|L|\ne \varnothing$.

From now on we will assume that X is a principally polarized abelian surface with real multiplication. We are interested in its Seshadri function

$$\begin{aligned} \varepsilon :\mathrm{Nef}(X)\rightarrow \mathbb {R},\quad L\mapsto \varepsilon (L)=\varepsilon (L,0) . \end{aligned}$$

Thanks to homogeneity, it is enough to consider this function on a compact cross-section of the nef cone. Any non-trivial nef class $L\in \mathrm{NS}_\mathbb {R}(X)$ is a positive multiple of a class of the form $L_t:=L_0+tL_\infty$ with suitable $t\in \mathbb {R}$. Applying Lemma 2.1, we see that if $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$, then the line bundle $L_t$ is nef if and only if $|t|\le \frac{1}{\sqrt{e}}$, and if $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$, then $L_t$ is nef if and only if $-{\textstyle \frac{2}{\sqrt{e}+1}}\le t\le {\textstyle \frac{2}{\sqrt{e}-1}}$. Ampleness holds when the inequalities are strict. We denote by $\mathcal N(X)=[-\frac{1}{\sqrt{e}},\frac{1}{\sqrt{e}}]$ and $\mathcal N(X)=[-{\textstyle \frac{2}{\sqrt{e}+1}}, {\textstyle \frac{2}{\sqrt{e}-1}}]$, respectively, the interval where $L_t$ is nef. This interval $\mathcal N(X)$ is a model for the cross-section of the nef cone, and therefore we will also write $L_t\in \mathcal N(X)$ instead of $t\in \mathcal N(X)$. So for every nef $\mathbb {R}$-line bundle, the ray $\mathbb {R}_{> 0}L$ has a unique representative in $\mathcal N(X)$, and we may consider the Seshadri function as

$$\begin{aligned} \varepsilon :\mathcal N(X)\rightarrow \mathbb {R},\quad t\mapsto \varepsilon (L_t). \end{aligned}$$

Any effective divisor D defines a linear function

$$\begin{aligned} \ell _D:\mathbb {R}\rightarrow \mathbb {R},\quad t\mapsto \frac{D\cdot L_t}{\mathrm{mult}_0 D} \end{aligned}$$

which computes the Seshadri quotient of the divisor D for any line bundle $L_t$. We denote the open subset containing all ample line bundles $L_t$, whose Seshadri quotient with D is submaximal, by $I_D$, i.e.,

and we call $I_D$ the submaximality interval of D.

It is a result of Szemberg [17, Prop. 1.8] that on any smooth projective surface S an ample line bundle can have at most $\rho (S)$ (two, in our case) submaximal curves at any given point. Using the restrictions derived from Propositions 1.3 and 1.4 we show that in many cases only one curve can exist:

Theorem 2.2

Let L be any ample line bundle on X with $\varepsilon (L)<\sqrt{L^2}$. Suppose that either

• $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$ for a non-square integer $e>0$, or

• $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$ for a non-square integer $e>0$, such that $e\equiv 1$ modulo 4 and e has a prime factor p with $p\equiv 5$ or 7 modulo 8,

holds. Then there exists exactly one irreducible curve C that is submaximal for L.

Proof

We will show that the restrictions given in Propositions 1.3 and 1.4 cannot hold for two submaximal curves. In fact, we will show that the following equations can never be satisfied by any two ample line bundles $L_1$ and $L_2$ and two positive integers $m_1$ and $m_2$:

• (i) $L_1^2=m_1^2-1$ for $m_1>1$ ,

• (ii) $L_2^2=m_2^2-1$ for $m_2>1$ ,

• (iii) $L_1\cdot L_2=m_1m_2$ .

Note that this also includes the case $C^2=m^2-4$ from Proposition 1.3, because in this case $\mathcal {O}_X(C)$ is an even multiple of another line bundle and dividing the equation by 4 leads to an equation of the form (i).

First we treat the more immediate case: $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$, where e is a non-square positive integer. Assume that there exist ample line bundles $L_1$ and $L_2$ satisfying (i) and (ii), with their numerical classes given by $L_i\equiv a_iL_0+b_iL_\infty \,$ for $i=1,2$. Then $m_1$ and $m_2$ must be odd, since $L_i^2$ is even. But the intersection number for any two line bundles on X is even,

$$\begin{aligned} (a_1L_0 + b_1L_\infty )\cdot (a_2L_0+b_2L_\infty )=2a_1a_2 -2eb_1b_2, \end{aligned}$$

and hence it can never equal $m_1m_2$.

Next we treat the more subtle case: $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$, where e is a non-square positive integer with $e\equiv 1$ modulo 4, which has a prime factor p with $p\equiv 5$ or 7 modulo 8. The crucial idea in this case is to consider the three equations modulo p. Assume that $L_1$ and $L_2$ are two line bundles satisfying (i)–(iii), with their numerical classes given by $L_i\equiv a_iL_0+b_iL_\infty \,$ for $i=1,2$. If we consider the equations

• (i) $2L_1^2=4a_1^2+4a_1b_1 +(1-e)b_1^2 = 2m_1^2-2$,

• (ii) $2L_2^2=4a_2^2+4a_2b_2 +(1-e)b_2^2 = 2m_2^2-2$,

• (iii) $2L_1\cdot L_2 = 4a_1a_2 + 2a_1b_2 + 2a_2b_1 + (1-e)b_1b_2 = 2m_1m_2$,

modulo p and replace $2a_i+b_i$ by $c_i$ for $i=1,2$, then the equations can be expressed by bilinear forms over the finite field $\mathbb {F}_{p}$. For (i) and (ii) we obtain

• (I) $\quad \left( \begin{array}{c} c_1 \\ m_1 \end{array} \right) ^T \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -2 \end{array} \right) \left( \begin{array}{c} c_1 \\ m_1 \end{array} \right) =c_1^2-2m_1^2=-2$,

• (II) $\quad \left( \begin{array}{c} c_2 \\ m_2 \end{array} \right) ^T \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -2 \end{array} \right) \left( \begin{array}{c} c_2 \\ m_2 \end{array} \right) =c_2^2-2m_2^2=-2$.

It follows that $(c_i,m_i)\ne (0,0)\in \mathbb {F}_{p}^2$. For equation (iii) we find that

• (III) $\quad \left( \begin{array}{c} c_1 \\ m_1 \end{array} \right) ^T \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -2 \end{array} \right) \left( \begin{array}{c} c_2 \\ m_2 \end{array} \right) =\left( \begin{array}{c} c_1 \\ -2m_1 \end{array} \right) ^T \cdot \left( \begin{array}{c} c_2 \\ m_2 \end{array} \right) =0$

and, therefore, we obtain

i.e., $c_2=-2m_1\lambda$ and $m_2=c_1\lambda$ for some $\lambda \in \mathbb {F}_{p}$. Using (I) and (II) we obtain

$$\begin{aligned} -2=\left( \begin{array}{c} c_2 \\ m_2 \end{array} \right) ^T \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -2 \end{array} \right) \left( \begin{array}{c} c_2 \\ m_2 \end{array} \right) = \lambda ^2 (4m_1^2-2c_1^2) =4\lambda ^2. \end{aligned}$$

This implies that $-2$ is a quadratic residue modulo p. But as $p\equiv 5$ or 7 modulo 8, this is impossible, and thus we arrive at a contradiction. $\square$

As a consequence of Theorem 2.2 we observe:

Corollary 2.3

Let $\mathrm{End}(X)$ be as in Theorem 2.2. Then for every ample $\mathbb {R}$-line bundle $L_\lambda$ with $\varepsilon (L_\lambda )<\sqrt{L_\lambda ^2}$ the Seshadri function is given by a linear function in a neighborhood of $L_\lambda$.

Proof

Let $L_\lambda$ be an ample $\mathbb {R}$-line bundle with $\varepsilon (L_\lambda )<\sqrt{L_\lambda ^2}$ and let C be any Seshadri curve of $L_\lambda$. By the previous Theorem 2.2 the curve C is the only submaximal curve for every $\mathbb {Q}$-line bundle in $I_C$. Assume now that there exists an ample $\mathbb {R}$-line bundle $L_t\in I_C$ with two submaximal curves. By continuity both curves remain submaximal in a neighborhood of $L_t$ and, thus, there also exist $\mathbb {Q}$-line bundles which also have two submaximal curves. This, however, is impossible by Theorem 2.2. $\square$

We will see that the assumption $\varepsilon (L_\lambda )<\sqrt{L_\lambda ^2}$ is essential for the validity of the statement in the corollary, and in fact we will show that the local behavior in the remaining case $\varepsilon (L_\lambda )=\sqrt{L_\lambda ^2}$ is surprisingly intricate (see Corollary 3.6).

Computer-assisted calculations suggest that Theorem 2.2 is in fact an “if and only if” statement, which means that in the remaining cases there should always exist a line bundle with two submaximal curves. In Sect. 5 we will show how the existence of a line bundle with two submaximal curves can be verified using computer-assisted calculations. Furthermore, we will provide a sequence of numbers $e_n$ with the property that there exists a line bundle with two submaximal curves on any abelian surface with $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}}+{\textstyle \frac{1}{2}}\sqrt{e_n}]$.

Before we continue studying the local behavior of the Seshadri function in the case where line bundles can have two submaximal curves, we prove a useful relation between submaximality intervals and reducibility of effective divisors:

Lemma 2.4

Let D be an effective divisor on X which is submaximal for some ample line bundle. If there exists another effective divisor $D'$ whose submaximality interval $I_{D'}$ satisfies $I_D\subsetneq I_{D'}$, then D is reducible.

Proof

Let $I_{D}=(a,b)$ and $I_{D'}=(c,d)$ be the submaximality intervals of D and $D'$, respectively. Denoting by F the general upper bound function $t\mapsto \sqrt{L_t^2}$, the linear function $\ell _D$ is given by the straight line joining the points (a, F(a)) and (b, F(b)) and, respectively, $\ell _{D'}$ by joining the points (c, F(c)) and (d, F(d)). Since F is strictly concave, the linear function $\ell _{D'}$ is strictly smaller than $\ell _{D}$ in $I_{D'}$. Therefore D can never compute the Seshadri constant for any line bundle. However, if D were irreducible, then D would compute its own Seshadri constant (see [4, Prop. 1.2]), which is a contradiction. $\square$

By [17, Prop. 1.8], the number of curves that can be submaximal for an individual ample line bundle $L_t$ is bounded. We will now show that the number remains bounded even when all line bundles in an open neighborhood of $L_t$ are considered, provided that $\varepsilon (L_t)<\sqrt{L_t^2}$. This is a consequence of the following lemma.

Lemma 2.5

Let D be an effective divisor on X which is submaximal for an ample line bundle $L_t$. Then there exists at most four irreducible curves which are submaximal for some line bundles in $I_D$.

Moreover, if D is irreducible, then there exists at most three irreducible curves which are submaximal for some line bundles in $I_D$.

Proof

Assume, there exists five pairwise distinct irreducible curves $C_1,\ldots ,C_5$ which are submaximal for some line bundles in $I_D=(a,b)$. Let $I_{C_i}=(a_i,b_i)$ be the submaximality interval of $C_i$ and let $L_{t_i}\in \mathcal N(X)$ be the unique representative of $\mathcal {O}_X(C_i)$. We will show that the submaximality interval of $C_3$ is contained in $I_D$, which by Lemma 2.4 would imply that $C_3$ is reducible.

Since $C_i$ is submaximal for some ample line bundle, $C_i$ is submaximal for $\mathcal {O}_X(C_i)$ by [4, Prop. 1.2]. Therefore, $C_i$ is submaximal for $L_{t_i}$ and, thus, $t_i\in I_{C_i}$. Moreover, since $C_i$ is the only submaximal curve for $\mathcal {O}_X(C_i)$, we have $t_i\notin I_{C_j}$ for $i\ne j$. By assuming $t_1<t_2<t_3<t_4<t_5$ we deduce for $i=2,3,4$ that

$$\begin{aligned} (a_i,b_i)\subset (t_{i-1},t_{i+1}) \quad \text{ and } \quad t_i\in (b_{i-1},a_{i+1})\,. \end{aligned}$$

(*)

The submaximality intervals $(a_1,b_1)$ and $(a_5,b_5)$ have to intersect with (a, b), because by assumption $C_1$ and $C_5$ are submaximal for some line bundles in $I_D$ and, thus, we have $a<b_1$ and $a_5<b$. Furthermore, $(*)$ implies that $t_2,t_3,t_4\in (b_1,a_5)$ and, as a consequence, the interval $(t_{2},t_{4})$ is contained in $(b_1,a_5)$ and, therefore, in $I_D$. Since $(a_3,b_3)$ is contained in $(t_{2},t_{4})$, it is also contained in $I_D$. This, however, implies that $C_3$ is reducible by Lemma 2.4, which is a contradiction.

For the second statement, we assume there exists three irreducible curves. Using the same notation and arguments as above, it follows that $t_2\in (b_1,a_3)$, $a<b_1$, and $a_3<b$. Hence, we have $t_2\in (b_1,a_3)\subset I_D$. But this means that $L_{t_2}$ has $C_2$ and D as submaximal curves, which is a contradiction. $\square$

Hence, we conclude for the local structure of the Seshadri function:

Corollary 2.6

For every ample $\mathbb {R}$-line bundle $L_t$ with $\varepsilon (L_t)<\sqrt{L_t^2}$ the Seshadri function is locally a piecewise linear function, i.e., it is locally the minimum of at most two linear functions.

As before, the assumption $\varepsilon (L_t)<\sqrt{L_t^2}$ is essential for this statement to be true (see Remark 3.8).

Clearly, if a line bundle L has two submaximal curves, then there exists a neighborhood of L such that every line bundle has two submaximal curves, since any submaximal curve will remain submaximal in a neighborhood of L. On the other hand, we show that every submaximal curve gives rise to an open interval, in which it is the only submaximal curve:

Proposition 2.7

Let $C\equiv qL_0+pL_\infty$ be an irreducible curve that is submaximal for some ample line bundle L on X. Then there exists a neighborhood U of $L_{\frac{p}{q}}$ in $\mathcal N(X)$ such that C is the only submaximal curve for all line bundles in U. In particular, the Seshadri function coincides with $\ell _C$ in U.

Proof

Since C is submaximal for some ample line bundle L, we know that C is also submaximal for $\mathcal {O}_X(C)$ by [4, Prop. 1.2], and in fact C is the only submaximal curve for $\mathcal {O}_X(C)$, since every $\mathcal {O}_X(C)$-submaximal curve has to be a component of C by [5, Lemma 5.2]. Thus, C is the only submaximal curve for $L_{\frac{p}{q}}$. Applying Lemma 2.5, there exist at most two other curves, which are submaximal for some line bundle $L'\in I_C$. Thus, the only possibility in which no such neighborhood of $L_\frac{p}{q}$ exists, is the case where one of the other curves $C'$ satisfies $\mathcal {O}_X(C)\cdot C'/\mathrm{mult}_0(C')=\sqrt{C^2}$. This, however, implies that $C'$ is a component of C by [5, Lemma 5.2]. $\square$

4 Seshadri function on abelian surfaces with real multiplication

In this section we will develop a method to algorithmically compute the Seshadri constant for any ample $\mathbb {Q}$-line bundle on X, proving Theorem C stated in the introduction. Furthermore, we will see that the local structure of the Seshadri function has unexpected behavior at $L_\lambda$ if $\varepsilon (L_\lambda )=\sqrt{L_\lambda ^2}$. Our strategy is to make use of Pell divisors in such a way that it is not necessary to explicitly know their multiplicity, but to use their expected multiplicity given by the Pell solution.

Definition 3.1

Let $L_\lambda$ be an ample $\mathbb {Q}$-line bundle with $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$ and let $q\in \mathbb {N}$ be the unique integer such that $qL_\lambda$ is a primitive $\mathbb {Z}$-line bundle, i.e., q is the denominator of a coprime representation of $\lambda =\frac{p}{q}$. Denote by (l, k) the primitive solution of the Pell equation $x^2-(qL_\lambda )^2 y^2=1$. We call

$$\begin{aligned} \pi _\lambda :\mathbb {R}\rightarrow \mathbb {R},\qquad t\mapsto \frac{kqL_\lambda \cdot L_t}{l} \end{aligned}$$

the Pell bound at $L_\lambda$, and

the submaximality interval of $\pi _\lambda$.

So if $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$ and P is a Pell divisor of $L_\lambda$, then we have the following chain of inequalities:

$$\begin{aligned} \varepsilon (L_\lambda ) \leqslant \frac{L_\lambda \cdot P}{\mathrm{mult}_0P} \leqslant \pi _\lambda (\lambda ) < \sqrt{L_\lambda ^2}. \end{aligned}$$

Moreover, $\pi _\lambda$ is an upper bound for the Seshadri function in the submaximality interval $J_\lambda$:

$$\begin{aligned} \varepsilon (L_t)\leqslant \frac{L_t\cdot P}{\mathrm{mult}_0P} \leqslant \pi _\lambda (t) < \sqrt{L_\lambda ^2} \qquad \text{ for } \text{ all }\, t\in J_\lambda . \end{aligned}$$

We will now establish two important connections between submaximal curves and Pell bounds. First we prove that every submaximal curve has a unique representative in the set of Pell bounds. Secondly, we will exhibit a relation between the submaximality interval of a Seshadri curve C of $L_\lambda$ and the submaximality interval of the Pell bound $\pi _\lambda$.

Proposition 3.2

Let $L_\lambda$ be an ample $\mathbb {Q}$-line bundle with $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$ and let $C\equiv qL_0+pL_\infty$ be an irreducible curve that is submaximal for some ample line bundle L on X. Then the following are equivalent:

• (i) Either C or 2C is the unique Pell divisor of $L_\lambda$.

• (ii) The linear functions $\ell _C$ and $\pi _\lambda$ coincide.

• (iii) We have $\lambda =\frac{p}{q}$.

Proof

The equivalence of (i) and (iii) is an immediate consequence of Proposition 1.3. Furthermore, the implication (i) $\Rightarrow$ (ii) also follows from Proposition 1.3, since it shows that the multiplicity of C coincides with the expected multiplicity given by the Pell solution, and therefore the linear functions $\ell _C$ and $\pi _\frac{p}{q}$ coincide.

For the implication (ii) $\Rightarrow$ (iii) we have to show that $\ell _C=\pi _\lambda$ implies $\lambda =\frac{p}{q}$. By Proposition 2.7 the linear function $\ell _C$ coincides with the Seshadri function in an open neighborhood U of $L_\frac{p}{q}$. For any Pell divisor P of $L_\lambda$ we have

$$\begin{aligned} \varepsilon (L_t)=\ell _C(t)\le \ell _P(t)\le \pi _\lambda (t)\quad \text{ for } \text{ all } \quad t\in U, \end{aligned}$$

and hence the linear function $\ell _P$ coincides with $\ell _C$, since by assumption $\pi _\lambda =\ell _C$.

We claim that for every component $C'$ of P the linear functions $\ell _{C'}$ and $\ell _C$ also coincide. For this, assume that there exists a $t_0\in U$ such that $\ell _C(t_0)<\ell _{C'}(t_0)$. Then, upon writing $P=C'+R$, we have

$$\begin{aligned} \frac{C\cdot L_{t_0}}{\mathrm{mult}_0 C}=\ell _C({t_0})=\ell _P({t_0})=\frac{(C'+R)\cdot L_{t_0}}{\mathrm{mult}_0 C' + \mathrm{mult}_0 R}. \end{aligned}$$

This, however, implies that

$$\begin{aligned} \frac{C\cdot L_{t_0}}{\mathrm{mult}_0 C}>\frac{R\cdot L_{t_0}}{\mathrm{mult}_0 R}, \end{aligned}$$

which is impossible, since C computes the Seshadri constant $\varepsilon (L_{t_0})$.

So we have shown that C and any component $C'$ of the Pell divisor P define the same linear function. This means, in particular, that any component $C'$ of P is also submaximal for the line bundle $\mathcal {O}_X(C)$. But $\mathcal {O}_X(C)$ has only C as a submaximal curve by [5, Lem. 5.2], and, therefore, $P=kC$ for $k\in \mathbb {N}$. This implies that $L_\frac{p}{q}$ and $L_\lambda$ are rational multiples of each other. But in $\mathcal N(X)$ this is only possible if $\lambda =\frac{p}{q}$. $\square$

Proposition 3.3

Let $L_\lambda$ be an ample $\mathbb {Q}$-line bundle with $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$ and let $J_\lambda =(t_1,t_2)$ be the submaximality interval of the Pell bound $\pi _\lambda$. Then every Seshadri curve C of $L_\lambda$ is submaximal on $(t_1,\lambda )$ or on $(\lambda ,t_2)$.

Proof

Assume that C is not submaximal on $(\lambda ,t_2)$, i.e., $\ell _C(t_2)>\sqrt{L_{t_2}^2}=\pi _\lambda (t_2)$. Furthermore, since C is a Seshadri curve of $L_\lambda$, we have $\ell _C(\lambda )\le \pi _\lambda (\lambda )$. Therefore the slopes $m_\lambda$ of $\pi _\lambda$ and $m_C$ of $\ell _C$ satisfy $m_\lambda < m_C$. But this implies that $\ell _C(t_1)<\pi _\lambda (t_1)$ and therefore C is submaximal on $(t_1,\lambda )$. $\square$

We will need the submaximality intervals of Pell bounds in the following explicit form:

Lemma 3.4

Let $L_\lambda$ be an ample $\mathbb {Q}$-line bundle with $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$, and let l, k and q be as in Definition 3.1. If $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$, then the submaximality interval $J_\lambda$ of $\pi _\lambda$ is given by

$$\begin{aligned} J_\lambda =\left( \frac{2ek^2q^2\lambda - l \sqrt{e}}{e(2k^2q^2+1)}, \frac{2ek^2q^2\lambda + l \sqrt{e}}{e(2k^2q^2+1)} \right) , \end{aligned}$$

and, if $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + \frac{1}{2} \sqrt{e}]$, then the submaximality interval $J_\lambda$ of $\pi _\lambda$ is given by

$$\begin{aligned} J_\lambda =\left( \frac{2+2ek^2q^2\lambda - 2l \sqrt{e}}{(e-1)+2eq^2k^2}, \frac{2+2ek^2q^2\lambda + 2l \sqrt{e}}{(e-1)+2eq^2k^2} \right) . \end{aligned}$$

Proof

The interval limits of the submaximality interval $J_\lambda =(t_1,t_2)$ are the solutions t of the equation

$$\begin{aligned} \sqrt{L_t^2}=\pi _\lambda (t). \end{aligned}$$

In the case $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$, the solutions are

$$\begin{aligned} t_{1,2}=\frac{2ek^2q^2\lambda \mp l \sqrt{e}\sqrt{l^2-(qL_\lambda )^2k^2}}{e(2ek^2q^2\lambda ^2+l^2)}. \end{aligned}$$

Upon applying the Pell equation $l^2-(qL_\lambda )^2 k^2=1$, these solutions can be expressed by

$$\begin{aligned} t_{1,2}=\frac{2ek^2q^2\lambda \mp l \sqrt{e}}{e(2k^2q^2+1)}. \end{aligned}$$

The case $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$ is computed analogously. $\square$

As the linear function $\ell _C$ of a submaximal curve C coincides with a Pell bound, the interval borders for Seshadri curves have the same structure. This reveals an interesting behavior of submaximality intervals of Seshadri curves:

Proposition 3.5

Let $C_1$ and $C_2$ be two submaximal curves on X. Then the submaximality intervals $I_{C_1}$ and $I_{C_2}$ are never adjacent to each other, i.e., if $I_{C_1}=(t_1,t_2)$ and $I_{C_2}=(s_1,s_2)$, then $t_1\ne s_2$ and $t_2\ne s_1$.

Proof

Using the computation of the interval limits of Lemma 3.4, it follows that the left-hand side of the interval is always of the form $a-b\sqrt{e}$ for some $a\in \mathbb {Q}$ and $b\in \mathbb {Q}^+$, whereas the right-hand side is of the form $a'+b'\sqrt{e}$ for some $a'\in \mathbb {Q}$ and $b'\in \mathbb {Q}^+$. Since 1 and $\sqrt{e}$ form a basis of the $\mathbb {Q}$-vector space $\mathbb {Q}(\sqrt{e})$, they can never coincide. $\square$

Corollary 3.6

Let $L_\lambda$ be any ample $\mathbb {R}$-line bundle such that $\varepsilon (L_\lambda )=\sqrt{L_\lambda ^2}$. For every neighborhood U of $\lambda$, the Seshadri function is the pointwise infimum of infinitely many linear functions $\pi _\mu$, but it is not a piecewise linear function on U.

Proof

In any neighborhood U of $\lambda$, the rational numbers $\mu \in U$ with $\varepsilon (L_\mu )<\sqrt{L_\mu ^2}$ are dense in U. Thus, by continuity of the Seshadri function we can express the Seshadri constant for any value $t\in U$ as an infimum of linear functions $\pi _\mu$. The only possibility for the Seshadri function to be a piecewise linear function in a neighborhood of $\lambda$ is, if the Seshadri function is computed near $\lambda$ by two linear functions $\ell _1$ and $\ell _2$ with $\ell _1(\lambda )=\ell _2(\lambda )=\sqrt{L_\lambda ^2}$. This, however, is impossible by Proposition 3.5. $\square$

By combining Corollaries 2.3 and 3.6 we deduce the following more general version of Theorem A stated in the introduction.

Theorem 3.7

Let $\mathrm{End}(X)$ be as in Theorem 2.2. Then the Seshadri function is broken linear.

Proof

It follows from Corollary 2.3 that for every point $t\in \mathcal N(X)$ with $\varepsilon (L_t)<\sqrt{L_t^2}$ the Seshadri function is a linear function in a neighborhood of t. By Proposition 3.5 the maximal intervals, on which the Seshadri function is linear are never adjacent to each other. Lastly, we have to argue that the set is nowhere dense and non empty. For this we consider ample line bundles of the form $L=qL_0+4qL_\infty$ for odd $q\in \mathbb {N}$ and $p\in \mathbb {Z}$. In this cases $L^2$ can never be a square number as $L^2\equiv 2\;(\mathrm{mod}~4)$. This yields a dense subset of lines bundles $L_{4q/p}$ in $\mathcal N(X)$ with $\varepsilon (L_{4p/q})<\sqrt{L_{4p/q}^2}$. As the Seshadri function is continuous, we get for each line bundle $L_{4q/p}$ an open neighborhood on which the Seshadri function is submaximal. Thus, M(X) is a nowhere dense subset of $\mathcal N(X)$. Explicit computations show that the Seshadri curve $C\in |4L_0|$ of $L_0$ with $\mathrm{mult}_0 C=6$ is not submaximal on $\mathcal N(X)$ and, therefore, the interval borders of the submaximality interval $I_C$ are contained in M(X). $\square$

Remark 3.8

Suppose that on X there is a line bundle $L_\lambda$ with two submaximal curves. Then there exists a neighborhood of $L_\lambda$ in which every line bundle has two submaximal curves, and thus there exist linear segments of the Seshadri function that are adjacent to each other. On the other hand, we have seen in Proposition 2.7 that there are also neighborhoods, in which only one submaximal curve exists. Furthermore, using Proposition 1.4 one can show that every line bundle in the submaximality interval $I_0$ of $L_0$ has only one submaximal curve C, which is the unique Pell divisor of $L_0$. Consequently, the limit points of this submaximal interval are accumulation points of (piecewise) linear segments. So in this case, as in the situation of Theorem 3.7, the Seshadri function does not consist of only finitely many linear pieces.

We return to the submaximality interval $J_\lambda =(t_1,t_2)$ of a Pell bound $\pi _\lambda$ by providing an upper bound for its length:

Lemma 3.9

Let $L_\lambda$ be an ample $\mathbb {Q}$-line bundle with $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$, and let l, k and q be as in Definition 3.1. Then the interval length of $J_\lambda =(t_1,t_2)$ is bounded by

$$\begin{aligned} t_2-t_1 <\frac{\sqrt{11}}{q\sqrt{e}}. \end{aligned}$$

Proof

Using the fact that the Pell equation is equivalent to

$$\begin{aligned} \frac{l}{kq}=\sqrt{L_\lambda ^2+\frac{1}{k^2q^2}}, \end{aligned}$$

the interval length can be determined via Lemma 3.4: In the case $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$ we get

$$\begin{aligned} t_2 - t_1 = \frac{2{\textstyle \frac{l}{kq}}}{\sqrt{e}(2kq+{\textstyle \frac{1}{kq}})} =\frac{2\sqrt{L_\lambda ^2+{\textstyle \frac{1}{k^2q^2}}}}{\sqrt{e}(2kq+{\textstyle \frac{1}{kq}})} =\frac{2\sqrt{2-\frac{2ep^2k^2-1}{k^2q^2}}}{\sqrt{e}(2kq+{\textstyle \frac{1}{kq}})} < \frac{\sqrt{2}}{q\sqrt{e}}, \end{aligned}$$

and in the case $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$ we obtain

$$\begin{aligned} t_2 - t_1 = \frac{4\frac{l}{kq} \sqrt{e}}{\frac{e-1}{kq}+2ekq}=\frac{4\sqrt{e}\sqrt{2+2\lambda -\frac{e-1}{2}\lambda ^2+\frac{1}{k^2q^2}} }{\frac{e-1}{kq}+2ekq}< \frac{2\sqrt{2+\frac{2}{e-1}+\frac{1}{k^2q^2}} }{q\sqrt{e}}<\frac{\sqrt{11}}{q\sqrt{e}}. \end{aligned}$$

In the third step we replaced the term $2\lambda - {\textstyle \frac{e-1}{2}}\lambda ^2$ with its maximum value $\frac{2}{e-1}$, and we use $e\ge 5$ in the last step. Also, we made use of the inequality ${\textstyle \frac{1}{k^2q^2}}\le {\textstyle \frac{1}{4}}$, which can be verified by explicitly considering all possible Pell solutions for $q=1$. $\square$

Remark 3.10

In the last step of the proof we could have used the more direct estimate ${\textstyle \frac{1}{k^2q^2}}\le 1$, which implies

$$\begin{aligned} \frac{2\sqrt{2+\frac{2}{e-1}+\frac{1}{k^2q^2}} }{q\sqrt{e}}<\frac{\sqrt{14}}{q\sqrt{e}}\, \end{aligned}$$

However, it turns out that this upper bound is not sufficient for our purposes (in particular, for the proof of Proposition 5.4).

The previous Lemma yields the following:

Corollary 3.11

For any given interval $I\subset \mathcal N(X)$ there exist only finitely many Pell bounds $\pi _\lambda$ that are submaximal on I.

Proof

Let s be the length of I and let $\pi _\lambda$ be a Pell bound that is submaximal on I. Then s is at most the length of $J_\lambda$, so that it follows from Lemma 3.9 that the denominator of $\lambda =\frac{p}{q}$ satisfies

$$\begin{aligned} s \le \frac{\sqrt{11}}{q\sqrt{e}}. \end{aligned}$$

Thus, $\lambda$ has to be contained in the finite set

$\square$

By combining Proposition 3.3 and Corollary 3.11 we will obtain a purely numerical method to compute the Seshadri constant and determine the Seshadri curves of $L_\lambda$:

Proposition 3.12

Let $L_\lambda$ be an ample $\mathbb {Q}$-line bundle with $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$, and $J_\lambda =(t_1,t_2)$ be the submaximality interval of $\pi _\lambda$. Let $s(\lambda )=\min \{\lambda -t_1,t_2-\lambda \}$, and consider the finite set . Then the Seshadri constant of $L_\lambda$ is given by

Moreover, every Seshadri curve C of $L_\lambda$ is represented by a unique Pell bound $\pi _\tau$ with $\tau \in A_\lambda$ and $\varepsilon (L_\lambda )=\pi _\tau (\lambda )$.

Proof

By Prop. 3.3 any Seshadri curve C of $L_\lambda$ is submaximal either on $(t_1,\lambda )$ or on $(\lambda , t_2)$, and therefore it is submaximal on an interval of length $s=\min \{\lambda -t_1,t_2-\lambda \}$. By Proposition 3.2 the linear function $\ell _C$ coincides with a unique Pell bound $\pi _\tau$, and therefore their submaximality intervals coincide. Thus, $\tau$ is an element of the finite set $A_\lambda$ by Corollary 3.11. $\square$

Furthermore, we can identify those Pell bounds which uniquely represent submaximal curves:

Proposition 3.13

Let $L_\lambda$ be an ample $\mathbb {Q}$-line bundle with $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$ and let $A_\lambda$ be as in Proposition 3.12. Then the following conditions are equivalent:

• (i) The Pell bound $\pi _\lambda$ coincides with $\ell _C$ for some submaximal irreducible curve C.

• (ii) Every Pell bound $\pi _\mu$ with $\mu \in A_\lambda \setminus \left\{ \lambda \right\}$ satisfies $\pi _\lambda (\lambda )<\pi _\mu (\lambda )$.

Proof

Assume that $\pi _\lambda =\ell _C$, and let $\pi _\mu$ be any Pell bound with $\mu \ne \lambda$. Proposition 3.2 shows that the unique representative of $\mathcal {O}_X(C)$ in $\mathcal N(X)$ is $L_\lambda$. As C computes the Seshadri constant of $L_\lambda$, we have $\pi _\lambda (\lambda )\le \pi _\mu (\lambda )$. Thus, we have to show that equality does not occur for $\lambda \ne \mu$.

Assume that $\pi _\lambda (\lambda )=\pi _\mu (\lambda )$ holds. We will show that this implies $\lambda =\mu$. By Proposition 2.7 the submaximal curve C computes the Seshadri constant in an open neighborhood U at $L_\lambda$, hence

$$\begin{aligned} \pi _\lambda (t)\le \pi _\mu (t)\quad \text{ for } \quad t\in U. \end{aligned}$$

This implies that the linear functions $\pi _{\lambda }$ and $\pi _\mu$ coincide, since otherwise we would have $\pi _{\lambda }(t)<\pi _\mu (t)$ for either $t<\lambda$ or $t>\lambda$, which is impossible because $\pi _{\lambda }$ computes the Seshadri function locally. But by Proposition 3.2 the linear function $\ell _C$ only coincides with the Pell bound $\pi _\lambda$ and, thus, we have $\lambda =\mu$.

For the other implication, we argue as in the proof of Proposition 3.12: For every Seshadri curve C of $L_\lambda$ there is a unique $\pi _\tau$ with $\ell _C=\pi _\tau$ and $\tau \in A_\lambda$. Since C computes the Seshadri constant of $L_\lambda$, the Pell bound $\pi _\tau$ computes the Seshadri constant in $\lambda$. In particular, we have $\pi _\tau (\lambda )\le \pi _\lambda (\lambda )$. Since by assumption $\pi _\lambda (\lambda )<\pi _\mu (\lambda )$ for $\mu \ne \lambda$, we conclude that $\tau =\lambda$, and therefore $\pi _\lambda =\ell _C$. $\square$

So far, the assumption $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$ was crucial for our arguments since they depended on the existence of Pell divisors. We will now show that the Seshadri constant can in fact be effectively computed for any ample $\mathbb {Q}$-line bundle. This will complete the proof of Theorem C stated in the introduction.

Theorem 3.14

There is an algorithm that computes the Seshadri constant of every given ample line bundle on principally polarized abelian surfaces with real multiplication.

Proof

If $L_\lambda$ is a $\mathbb {Q}$-line bundle such that $\sqrt{L_\lambda ^2}\notin \mathbb {Q}$, then the assertion follows from the fact, that the set $A_\lambda$ from Proposition 3.12 is finite. Suppose then that $\sqrt{L_\lambda ^2}\in \mathbb {Q}$. We will construct a theoretical interval around $L_\lambda$, on which every Seshadri curve of $L_\lambda$ must be submaximal, if $\varepsilon (L_\lambda )<\sqrt{L_\lambda ^2}$. By Corollary 3.11, there are only finitely many Pell bounds on this interval, and $\varepsilon (L_\lambda )$ is the minimum of those.

Assume that the Seshadri constant satisfies $\varepsilon (L_\lambda )<\sqrt{L_\lambda ^2}$. Let $\lambda =\frac{p}{q}$ be a coprime representation. Then, $L:=qL_\lambda$ is a primitive $\mathbb {Z}$-line bundle. Denote by C any Seshadri curve of $L_\lambda$. As explained in Sect. 1, there exists an effective Divisor $D\in |2L|^+$ such that D satisfies ${D\cdot L}/{\mathrm{mult}_0(D)}\le \sqrt{L^2}$. As C is the Seshadri curve of L, C is a component of D by [5, Lemma 6.2]. It follows that the intersection-number $C\cdot L$ is bounded by $D\cdot L=2L^2$. As a consequence, the Seshadri constant can only take certain rational values:

Therefore, we find that the Seshadri constant is at most

$$\begin{aligned} \varepsilon (L)\le \frac{2L^2-1}{2\sqrt{L^2}}. \end{aligned}$$

For the construction of the interval, we will give a lower and an upper bound for the slope of the linear function $\ell _C$. To this end, we will chose any two rational numbers $\mu _i\in \mathcal N(X)$ with $\mu _1<\lambda <\mu _2$ and $\sqrt{L_{\mu _i}^2}\notin \mathbb {Q}$ for $i=1,2$. Next, we compute a Seshadri curve $C_i$ of $L_{\mu _i}$ using Proposition 3.13. We denote by $m_i$ the slope of the linear function $\ell _{C_i}$. As the Seshadri function is a concave function, the slope m of the linear function $\ell _C$ is bounded, $m_2\le m\le m_1$. Let $r_i$ be the linear function passing through the point $(\lambda ,({2L^2-1})/({2\sqrt{L^2}}))$ with slope $m_i$. Then the function $u(t)=\max \left\{ r_1(t),r_2(t)\right\}$ is an upper bound for $\ell _C$, since we have

$$\begin{aligned} \ell _C(t) \le r_1(t) \quad \text{ for } \lambda \le t \quad \text{ and } \quad \ell _C(s) \le r_2(s) \quad \text{ for } \quad s\le \lambda . \end{aligned}$$

Denote by I the submaximality interval of u. It follows that C has to be submaximal on I, and so we have constructed a computable interval on which C is submaximal. Now, by following the same argument from Proposition 3.12 we can compute the Seshadri constant of $L_\lambda$ by taking the minimum of Pell bounds in $\lambda$, which are submaximal on I. Clearly, if none of these Pell bounds are submaximal in $\lambda$, then the Seshadri constant satisfies $\varepsilon (L_\lambda )=\sqrt{L_\lambda ^2}$. $\square$

Remark 3.15

In the proof of Theorem 3.14 we have shown how to algorithmically distinguish the cases $\varepsilon (L_\lambda ^2)<\sqrt{L_\lambda }$ and $\varepsilon (L_\lambda )=\sqrt{L_\lambda ^2}$. Both cases do, in fact, occur for line bundles L with $\sqrt{L^2}\in \mathbb {Q}$: Consider a principally polarized abelian surface X with $\mathrm{End}(X)=\mathbb {Z}[\sqrt{2}]$. Then the line bundle $L=2L_0+L_\infty$ satisfies $\varepsilon (L)=\sqrt{L^2}=2$, whereas the line bundle $L'=58L_0+L_\infty$ satisfies $\varepsilon (L')<\sqrt{L'^2}$, since the Seshadri curve C of $L_0$ is also submaximal for $L'$.

It is an important consequence of Theorem 3.14 that the Seshadri function depends only on the endomorphism ring of X, but not on the isomorphism class of the surface:

Theorem 3.16

Let X and Y be (not necessarily isomorphic) principally polarized abelian surfaces with real multiplication with $\mathrm{End}(X)\cong \mathrm{End}(Y)$. Then their Seshadri functions coincide in the following sense: Choosing suitable bases of the Néron-Severi groups $\mathrm{NS}(X)$ and $\mathrm{NS}(Y)$ yields an isomorphism $\mathrm{Nef}(X)\simeq \mathrm{Nef}(Y)$, under which we have $\varepsilon _X=\varepsilon _Y$.

This implies Corollary D stated in the introduction.

Proof

The proof of Theorem 3.14 shows that the numerical data that enters the computation of the Seshadri functions stems from the endomorphism ring. Therefore, an isometry of $\mathrm{NS}(X)$ that leaves the ample cone invariant also leaves the Seshadri function invariant. $\square$

5 Fundamental cone and sample plots for Seshadri functions

We will now determine the subgroup $G\subset \mathrm{Aut}(\mathrm{NS}(X))$ of isometries with respect to the intersection product that leave the Seshadri function on $\mathrm{Amp}(X)$ invariant. This group gives rise to a decomposition of the ample cone into subcones on which G acts transitively.

With respect to the basis $(L_0,L_\infty )$, an automorphism $\varphi \in \mathrm{Aut}(\mathrm{NS}(X))$ is given by a matrix

By Theorem 3.16 the Seshadri function remains invariant under the automorphism $M_\varphi$ if it is an isometry of $\mathrm{NS}(X)$ and additionally leaves the ample cone invariant. These conditions can be expressed by:

• (i) $L_0^2=(\alpha L_0 + \gamma L_\infty )^2$,

• (ii) $L_\infty ^2=(\beta L_0 + \delta L_\infty )^2$,

• (iii) $L_0\cdot L_\infty =(\alpha L_0 + \gamma L_\infty )\cdot (\beta L_0 + \delta L_\infty )$,

• (iv) $\alpha > 0$.

The conditions (i)–(iii) are equivalent to $\varphi$ being an isometry, whereas condition (vi) ensures that the ample cone is left invariant.

In the case of $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$ we find by solving (i)–(iv) that $M_\varphi$ is of the form

Since any other Pell solution of $x^2-ey^2=1$ is generated by the minimal solution $(\alpha _0,\beta _0)$, the group G is generated by

A line bundle $L=aL_0+bL_\infty$ is a principal polarization if and only if (a, b) is a solution of Pell’s equations $x^2-ey^2=1$ with $a>0$. Therefore, we can express every principal polarization by $L_k:=x_kL_0+y_kL_\infty$, where $(x_k,y_k)$ satisfies

So we have $\varphi _0(L_k)=L_{k+1}$. Next, we consider for $k\in \mathbb {Z}$ the subcone $\mathcal {D}_k\subset \mathrm{Amp}(X)$ generated by $L_k$ and $L_{k+1}$. We have $\varphi _0^k(\mathcal {D}_0)=\mathcal {D}_k$. Additionally, by also considering the automorphism $\tau$ we can further divide the subcone $\mathcal {D}_0$ into two subcones $\mathcal {D}_{0,1}$ and $\mathcal {D}_{0,2}$ as follows: The automorphism $\varphi _0\circ \tau$ is of order two and maps the cone $\mathcal {D}_0$ onto itself. The line bundle $L':=e\beta _0L_0+(\alpha _0-1)L_\infty$, which satisfies $\varphi _0\circ \tau (L')=L'$, divides the subcone $\mathcal {D}_0$ into two subcones $\mathcal {D}_{0,1}$ and $\mathcal {D}_{0,2}$, where $\mathcal {D}_{0,1}$ is generated by $L_0$ and $L'$,

and $\mathcal {D}_{0,2}$ is generated by $L'$ and $L_1$. The subcones $\mathcal {D}_{0,1}$ and $\mathcal {D}_{0,2}$ satisfy $\varphi _0\circ \tau (\mathcal {D}_{0,1})=\mathcal {D}_{0,2}$ and, again by construction of $\tau$, the Seshadri constants remain invariant. We call $\mathcal {D}_{0,1}$ the fundamental cone of $\mathrm{Amp}(X)$. This cone corresponds to the interval $[0,{\textstyle \frac{\alpha _0-1}{e\beta _0}}]$ in $\mathcal N(X)$. The decomposition of the subcone $\mathcal {D}_0$ into $\mathcal {D}_{0,1}$ and $\mathcal {D}_{0,2}$ extends via $\varphi _0^k$ to every subcone $\mathcal {D}_k$. After renumbering we obtain a decomposition of $\mathrm{Amp}(X)$ into subcones $\mathcal {C}_k$ with $\mathcal {C}_0=\mathcal {D}_{0,1}$.

We now deal with the case of $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$. In this case we find that $M_\varphi$ is given by

Note that we have the following bijection

By [8, Prop. 6.3.16] the set of solutions for the Pell-type equation $x^2-ey^2=4$ can be expressed through a minimal solution $(x_0,y_0)$ (we may assume $x_0>y_0>0$) as follows:

With some calculation, the set of solutions of $\alpha ^2+\alpha \beta -{\textstyle \frac{e-1}{4}}\beta ^2=1$ can be determined as

where $(\alpha _0,\beta _0):=({\textstyle \frac{x_0-y_0}{2}},y_0)$ and, hence, the group G is generated by

Using the exact same argument as before, we get a decomposition of the ample cone: We can express every principal polarization by $L_k:=x_kL_0+y_kL_\infty$ with $(x_k,y_k)=\psi _0^k(1,0)$. The subcones $\mathcal {D}_k$ generated by $L_k$ and $L_{k+1}$ satisfy $\psi _0^k(\mathcal {D}_0)=\mathcal {D}_k$. Furthermore, $\psi _0\circ \sigma$ divides the subcone $\mathcal {D}_0$ into two subcones $\mathcal {D}_{0,1}$ and $\mathcal {D}_{0,2}$, which are generated by $L_0$ and $L':=(\alpha _0+1)L_0+\beta _0L_\infty$ and, respectively, $L'$ and $L_1$. In this case, the fundamental cone $\mathcal {D}_{0,1}$ corresponds to the interval $[0,{\textstyle \frac{\beta _0}{\alpha _0+1}}]$ in $\mathcal N(X)$.

The considerations above prove Theorem B stated in the introduction.

We now provide some sample plots in order to illustrate the behavior of Seshadri functions. Concretely, we compute for fixed e all Seshadri curves $C=qL_0+pL_\infty$ with $q\le 3.000$ that are contained in $\mathcal {C}_0$. From this set of curves we derive further Seshadri curves by applying the automorphisms in G. In the pictures, the dotted lines indicate the fundamental interval from which the complete Seshadri function can be computed by Theorem B.

The values of e in Figs. 1, 2, 3 and 4 are chosen in such a way that they illustrate different kinds of behavior: In the case of $\mathbb {Z}[\sqrt{2}]$ there exist $\mathbb {Q}$-line bundles $L_\lambda$ with $\varepsilon (L_\lambda )=\sqrt{L_\lambda ^2}\in \mathbb {Q}$ whereas in the case of $\mathbb {Z}[\sqrt{5}]$ no such bundles exist. These line bundles generate “gaps” in the graph, because they do not give rise to a linear segment. In fact, at each of these gaps there are infinitely many linear segments which converge from both sides. In the plots for the case $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}}+{\textstyle \frac{1}{2}} \sqrt{e}]$ for $e=5$ and 33 the ample cone is not symmetric at 0 in the case of $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}}+{\textstyle \frac{1}{2}} \sqrt{e}]$.

Fig. 1

The Seshadri function of an abelian surface with real multiplication in $\mathbb {Z}[\sqrt{2}]$

Fig. 2

The Seshadri function of an abelian surface with real multiplication in $\mathbb {Z}[\sqrt{5}]$

Fig. 3

The Seshadri function of an abelian surface with real multiplication in $\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}}\sqrt{5}]$

Fig. 4

The Seshadri function of an abelian surface with real multiplication in $\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}}\sqrt{33}]$

The Seshadri function for $e=5$ consists only of linear segments which by Theorem 2.2 are never adjacent to each other. In the case of $e=33$ there exist line bundles with two submaximal curves, e.g., at $t=0.37$. In fact, calculations show that there are chains of linear segments which overlap. It should also be noted that the size of the fundamental interval depends heavily on the minimal solution of $x^2-ey^2=1$ or, respectively, $x^2+xy-{\textstyle \frac{e-1}{4}}x^2=1$: In the first three cases the minimal solutions are small, which leads to a small fundamental interval. However, experience with further examples has shown that the limit of the fundamental interval can be arbitrarily close to the interval limit of $\mathcal N(X)$.

6 Distinguishing the cases of one and two submaximal curves

In this section we derive a method that allows one to distinguish whether all line bundles on X have at most one submaximal curve or if there exists a line bundle which has 2 submaximal curves. By Theorem 2.2 we already know that there are infinitely many cases for $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$, where every line bundle has at most one submaximal curve. We will show that the case with two submaximal curves also appears infinitely many times.

Proposition 5.1

There exists a line bundle on X that has two submaximal curves if and only if there exist two Pell bounds $\pi _\lambda$ and $\pi _\mu$ such that the following two conditions are met:

• (i) Their submaximality intervals $J_\lambda$ and $J_\mu$ intersect and one is not contained in the other.

• (ii) There does not exist a Pell bound $\pi _\tau$ such that the submaximality interval $J_\tau$ contains $J_\lambda \cup J_\mu$.

Proof

Assume that there exists a line bundle L on X with two submaximal curves $C_1$ and $C_2$. The linear functions $\ell _{C_1}$ and $\ell _{C_2}$ are Pell bounds by Proposition 3.2, and their submaximality intervals $I_{C_1}$ and $I_{C_2}$ must intersect, because $C_1$ and $C_2$ are both L-submaximal. By Lemma 2.4 one submaximality interval can not be contained in the other. Assume that there exists a Pell bound $\pi _\tau$ such that $I_{C_1}\cap I_{C_2} \subset J_\tau$. Then any Pell divisor P of $L_\tau$ is submaximal on $I_{C_1}\cap I_{C_2}$, and therefore $C_1$ and $C_2$ are reducible by Lemma 2.4, a contradiction.

Suppose now that there exist two Pell bounds $\pi _\lambda$ and $\pi _\mu$ such that (i) and (ii) holds. The Pell bounds yields an upper bound for the Seshadri function in $J_\lambda \cup J_\mu$: We have

$$\begin{aligned} \varepsilon (t)\le \min \{\pi _\lambda (t),\pi _\mu (t)\}< \sqrt{L_t^2} \qquad \text{ for } t\in J_\lambda \cup J_\mu \text{. } \end{aligned}$$

Let $C_1$ be a Seshadri curve for a line bundle $L_{t_1}$ with $t_1\in J_\lambda \cup J_\mu$. Due to (ii) the submaximality interval $I_{C_1}$ of $C_1$ cannot cover the complete interval $J_\lambda \cup J_\mu$. Therefore, by continuity there exists a $t_2\in (J_\lambda \cup J_\mu ) \cap I_{C_1}$ such that

$$\begin{aligned} \varepsilon (L_{t_2})\le \min \{\pi _\lambda (t_2),\pi _\mu (t_2)\}< \frac{C_1\cdot L_{t_2}}{\mathrm{mult}_0 C_1} < \sqrt{L_{t_2}^2}\,, \end{aligned}$$

i.e., $C_1$ is submaximal for $L_{t_2}$ but does not compute its Seshadri constant. But the Seshadri constant of $L_{t_2}$ is computed by a curve, and thus there exists for $L_{t_2}$ another submaximal curve $C_2$ that computes the Seshadri constant. It follows that $L_{t_2}$ has two submaximal curves. $\square$

The criterion in Proposition 5.1 provides us with a numerical method to search for line bundles with two submaximal curves: First, we search for Pell bounds whose submaximality intervals intersect. After that, one checks by using Proposition 3.12 whether there exists another Pell bound which contains both intervals. Using computer-assisted computation, this yields the following:

Proposition 5.2

Suppose that $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$ for a non-square integer e with $0<e\leqslant 25.000$, such that we have $e\equiv 1$ modulo 4 and e does not have a prime factor p with $p\equiv 5$ or 7 modulo 8. Then there exists a line bundle on X with two submaximal curves.

Theorem 2.2 and the previous proposition suggest the following conjecture:

Conjecture 1

Let L be any ample $\mathbb {Q}$-line bundle on X. Then there exists at most one irreducible curve C that is submaximal for L if and only if $\mathrm{End}(X)$ satisfies either

• $\mathrm{End}(X)=\mathbb {Z}[\sqrt{e}]$ for a non-square integer $e>0$, or

• $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}} \sqrt{e}]$ for a non-square integer $e>0$, such that $e\equiv 1$ modulo 4 and e has a prime factor p with $p\equiv 5$ or 7 modulo 8.

Remark 5.3

One can show by applying well-known results on quadratic residues and binary quadratic forms (see e.g. [8, Prop. 2.2.4] and [9, Lemma 2.5]) that the following conditions are equivalent for a non-square integer e with $e\equiv 1$ modulo 4:

• (i) e does not have any prime factor p with $p\equiv 5$ or 7 modulo 8.

• (ii) $-2$ is a quadratic residue modulo e.

• (iii) $e=A^2+8B^2$ for some $A,B\in \mathbb {N}$ with $\gcd (A,B)=1$.

Finally, we will show that the case with two submaximal curves occurs infinitely often.

Proposition 5.4

Let $e_n:=1+8n^2$. If $e_n$ is not a perfect square, then every principally polarized abelian surface with $\mathrm{End}(X)=\mathbb {Z}[{\textstyle \frac{1}{2}} + {\textstyle \frac{1}{2}}\sqrt{e_n}]$ has a line bundle with two submaximal curves.

Proof

Consider the ample line bundles $L=2nL_0+L_\infty$ and $L'=(2n-1)L_0+L_\infty$. The Pell solution of $x^2-L^2y^2=1$ is given by $(2n+1,1)$, and the Pell solution of $x^2-L'^2y^2=1$ is $(2n-1,1)$. Hence, the submaximality intervals of the corresponding Pell bounds $\pi _\frac{1}{2n}$ and $\pi _\frac{1}{2n-1}$ are given by

$$\begin{aligned} J_\frac{1}{2n}=\left( \frac{16n^3+(2n+1) (1-\sqrt{8n^2+1})}{8n^2(4n^2+1)},\frac{16n^3+(2n+1)(1+\sqrt{8n^2+1})}{8n^2(4n^2+1)}\right) \end{aligned}$$

and, respectively,

$$\begin{aligned} J_\frac{1}{2n-1}=\left( \frac{2n+(2n-1)(8n^2-\sqrt{8n^2+1})}{32n^4-32n^3+16n^2-4n+1},\frac{2n+(2n-1)(8n^2+\sqrt{8n^2+1})}{32n^4-32n^3+16n^2-4n+1} \right) \,. \end{aligned}$$

Explicit computations show that both Pell bounds $\pi _\frac{1}{2n}$ and $\pi _\frac{1}{2n-1}$ are submaximal at $\frac{2}{4n-1}$, and therefore their submaximality intervals intersect.

So the first condition of Proposition 5.1 is satisfied. In order to conclude that a line bundle with two submaximal curves exists, it remains to show that there does not exist another Pell bound $\pi _\lambda$ whose submaximality interval covers the interval $I := J_\frac{1}{2n}\cup J_\frac{1}{2n-1}$. For this, we will derive an upper bound and a lower bound for the denominator q of $\lambda =\frac{p}{q}$ which must be satisfied if the Pell bound $\pi _\lambda$ covers I. As we will see, the upper and lower bound contradict each other and thus there cannot exist such a Pell bound.

Upper bound for q: The Pell bound $\pi _\lambda$ has to cover both submaximality intervals, i.e., the interval

$$\begin{aligned} I=\left( \frac{16n^3+(2n+1)(1 -\sqrt{8n^2+1})}{8n^2(4n^2+1)},\frac{2n+(2n-1)(8n^2+\sqrt{8n^2+1})}{32n^4-32n^3+16n^2-4n+1} \right) . \end{aligned}$$

One can show that the length of this interval is at least $(\sqrt{2}+1)/(4n^2)$, and using Lemma 3.9 we derive the upper bound

$$\begin{aligned} q\le \frac{4\sqrt{11}n^2}{(\sqrt{2}+1)\sqrt{8n^2+1}}\le \frac{35}{18}n. \end{aligned}$$

Lower bound for q: First, we observe that the unique Pell bound $\pi _0$ is not submaximal for L and $L'$ and, thus, we may assume that $\lambda \ne 0$, i.e. $p\ne 0$. We obtain a preliminary lower bound for q by taking into account that the line bundle $L_\lambda$ has to be ample, i.e.,

$$\begin{aligned} L_\lambda ^2=2+\frac{2p}{q} - \frac{4n^2p^2}{q^2} >0, \end{aligned}$$

and, thus,

$$\begin{aligned} q\ge \frac{p}{2}(\sqrt{8n^2+1}-1)\ge \sqrt{2}\,(n-1). \end{aligned}$$

Unfortunately, this lower bound yields no contradiction with our upper bound. However, it provides us with a method to refine the lower bound. Using the computation from Lemma 3.9, we find a maximal possible length for the submaximality interval $J_\lambda =(t_1,t_2)$ provided that $\sqrt{2}\,(n-1)\le q \le \frac{35}{18}n$:

$$\begin{aligned} t_2 - t_1< \frac{2\sqrt{2+\frac{2}{e_n-1}+\frac{1}{k^2q^2}} }{q\sqrt{e_n}}\le \frac{2\sqrt{2+\frac{1}{4n^2}+{\textstyle \frac{1}{2\,(n-1)^2}}} }{\sqrt{2}(n-1)\sqrt{1+8n^2}} < \frac{\sqrt{2+\frac{1}{4n^2}+{\textstyle \frac{1}{2\,(n-1)^2}}} }{2n(n-1)}. \end{aligned}$$

This in turn, gives us an upper bound for $\lambda$, since the submaximality interval of $\pi _\lambda$ can cover at most $t_1-t_2$:

$$\begin{aligned} \lambda \le \frac{16n^3+2n+1 -(2n+1)\sqrt{8n^2+1}}{8n^2(4n^2+1)} + \frac{\sqrt{2+\frac{1}{4n^2}+{\textstyle \frac{1}{2\,(n-1)^2}}} }{2n(n-1)}. \end{aligned}$$

It follows that $\lambda \le \frac{1}{2n-3}$ and, therefore, the denominator q of $\lambda$ must be at least $2n-3$.

This shows that for $n\ge 55$ there cannot exist a Pell bound whose submaximality interval covers $J_\frac{1}{2n}$ and $J_\frac{1}{2n-1}$. Thus, the assertion follows for $n\ge 55$ from Proposition 5.1. The explicit computations from Proposition 5.2 cover the remaining cases for $n\le 54$. $\square$

Remark 5.5

The case where $e_n$ is a square number, i.e., $e_n=1+8n^2=r^2$ for an integer $r\in \mathbb {N}$, is equivalent to the case where (r, n) is a solution for the Pell equation $x^2-8y^2=1$, and hence there are infinitely many n such that $e_n$ is not a square number.