Curves with many points Noam D. Elkies

Curves with many points Noam D. Elkies Introduction. Let C be a (smooth, projective, absolutely irreducible) curve of genus g 2 over a number field K. Faltings [Fa1, Fa2] proved that the set C(K) of K-rational points of C is finite, as conjectured by Mordell. The proof can even yield an effective upper bound on the size #C(K) of this set (though not, in general, a provably complete list of points); but this bound depends on the arithmetic of C. This suggests the question of how #C(K) behaves as C varies. Following [CHM1, CHM2], given g > 2 and K we define: B(g, K) = max C #C(K), (1) with C running over all curves over K of genus g; N(g, K) = lim sup C #C(K) [ B(g, K)], (2) (so infinitely many C have N rational points over K, but only finitely many have more than N), and N(g) = max K N(g, K). (3) Note that it is essential to use N(g, K) here rather than B(g, K), because even for a single curve C over a number field K 0 it is clear that by enlarging K K 0 we can make #C(K) arbitrarily large. It is far from obvious that either B(g, K) or N(g) is finite for any g, K; even the question of whether N(2, Q) < is very much open. But Caporaso, Harris and Mazur recently proved [CHM1] that Lang s Diophantine conjectures [La] imply the finiteness of B(g, K), N(g) for any number field K and integer g 2. More precisely, Lang conjectured that if V/K is a variety of general type then: i) V (K) is not Zariski dense in V, and ii) V 0 V closed, proper such that for all number fields K K. V (K ) = V 0 (K ) (finite) (4) Caporaso, Harris and Mazur prove: Conj. (i) for some number field K implies the finiteness of B(g, K) for all g 2, and Conj. (ii) N(g) < for all g 2. 1

This raises new questions: What are those B s and N s, and how fast do they grow with g? Even assuming Lang s conjectures it still seems hopeless to obtain any upper bounds. However, lower bounds require only the construction of curves or families of curves C with #C(K) large. We thus come to our main theme: How to construct curves of given genus g > 1 with many rational points? All constructions so far follow one of two main strategies. To estimate N(g) or N(g, K) from below, we need infinitely many curves C, and always 1 obtain them from a family of curves of genus g, i.e. a variety C with a map to some base B/K whose generic fiber is a curve of genus g, together with N sections σ i : B C defined over K. Our curves C will then be fibers of C above K- rational points of B. (This approach may be regarded as the flip side of the proof in [CHM1] that N(g) is bounded assuming Lang s conjectures.) These specializations of C also yield lower bounds on B(g, K), but particularly for small g and K = Q one might find a specialization C with further rational points not accounted for by the sections σ i, or even an isolated curve that happens to have many points but does not generalize to any known nice family C. In this paper we concentrate on the first, geometric, strategy, and in particular on lower bounds for N(g). We begin with a brief discussion of the requirements that the family C B must satisfy to be of use to us, and then tabulate the best N(g) bounds known and exhibit the known families with many sections. For all but finitely many g, the current (May 1996) best lower bound on N(g) is 16(g + 1), due independently to Brumer and Mestre. Mestre s approach also yields the best lower bounds known for N(g, Q) with g large. We present these methods first. We then present two alternative approaches suggested by J. Harris, and pursue them to improve on the Brumer-Mestre bound for certain small g, notably 2, 3, 4, 5, 6. Finally we quote findings of Kulesz, Keller and Stahlke on individual curves of low genus with many Q-rational points. Families of curves. Let C B be a family of curves of genus g over the number field K. Since we want infinitely many specializations, we require that B have infinitely many K-rational points, and may as well assume that 1 At least, all constructions so far fit this geometric description, though there could conceivably be other approaches. 2

B is irreducieble and B(K) is Zariski ense in B, else we could replace B by a positive-dimensional component of the Zariski closure of B(K). (In all such constructions so far B is either a rational variety of positive dimension or an elliptic curve of positive rank.) Since the points parametrizing degenerate curves, or curves where σ i = σ j for some i j, fall on a subvariety of positive codimension, we then obtain infinitely many specializations to a curve of genus g with at least N distinct rational points. To conclude that N(g, K) > N we must also ensure that these specializations are in infinitely many K-isomorphism classes. Lower bounds on N(g). In the following table, BM is the Brumer- Mestre 8g + 16 bound, L denotes curves obtained by slicing surfaces in P 3 with many Lines, T indicates Twists of a fixed curve with many symmetries, and F means a nonisotrivial Family of highly symmetric curves. g 2 3 4 5 6 7 8 9 10 > 10 (but 45) 45 N(g) 128 100 126 132 146 128 144 180 192 16(g + 1) 781 method # L T F T L BM BM L T BM L 2. Brumer and Mestre s curves For all g > 6 except 9,10,45 the record is held by Mestre and Brumer, who independently get N(g) 16(g + 1). Though they approach the problem differently, Brumer later showed that his curves turn out to include Mestre s as a special case. Mestre s approach also yields the record for N(g, Q) for large g. All these curves are hyperelliptic, i.e. of the form y 2 = P (x) for some polynomial P with distinct roots and degree m = 2g + 2. Brumer s curves are of the form C A,B : y 2 = Ax m + B (5) with AB 0 over a field K containing the m-th roots of unity, and C A,B : y 2 = Ax m + Bx m/2 + A (6) with A 0, B ±2A, and K containing the roots of unity of order m/2. Both curves have 2m automorphisms: on C A,B we may multiply y by ±1 and x by any m-th roots of unity; on C A,B we have (x, y) (ζx, ±y) with ζ m/2 = 1, and also (x, y) (1/x, y/x m/2 ). These automorphisms descend to a cyclic or dihedral group G PGL 2 (K) of size m acting on the projective x-line. Now choose any x 1, x 2, x 3, x 4 in distinct orbits of G, such that each of the x i has trivial stabilizer in G (this excludes only finitely many points 3

of P 1 ). Suppose we can choose A, B so that C A,B has a K-rational point above each x i, i.e. such that u i A + v i B = y 2 i (i = 1, 2, 3, 4) (7) for some y i K, where (u i, v i ) = (x m i, 1) for C A,B and (x m i + 1, x m/2 i ) for C A,B. Then, using the 2m automorphisms of that curve, we find at least 4 2m = 16(g + 1) rational points. But (7) is a homogeneous equation for an elliptic curve, which has positive rank at least if we further enlarge K. Mestre obtains curves C A,B with at least 8m points over K µ m by adapting one of his constructions of elliptic curves with large rank. The main tool is the case r = 2 of the following Lemma M. Let r, d be positive integers with r > 1, and let P be a monic polynomial of degree rd over a field K of characteristic zero or prime to r. Then there are unique polynomials Q, R such that P = Q r R where Q is monic of degree d and R has degree < (r 1)d. Moreover, for generic monic P of degree rd, the remainder R has degree exactly (r 1)d 1. Proof : Expand P = Q r R in a Laurent series about and take r-th roots to find that P 1/r Q = o(1), i.e. Q is the polynomial part of the Laurent expansion of P 1/r. Conversely, every choice of Q, R with Q monic of degree d and R of degree < (r 1)d makes Q r R monic of degree rd. Thus a generic choice of P is equivalent to a generic choie of Q, R, and the latter choice makes R a polynomial of degree (r 1)d 1, as claimed. This lemma is, of course, well-known and elementary, but it was Mestre who first used it systematically in this context. In our case, r = d = 2, and P is the polynomial P (X) = (X x m 1 )(X x m 2 )(X x m 3 )(X x m 4 ). (8) Lemma M yields P = Q 2 R with Q quadratic and R linear. Since P (x m i ) = 0 for i = 1, 2, 3, 4 it follows that the curve y 2 = R(x m ) has rational points (x i, Q(x i )). This is a curve of the form C A,B parametrized by (x 1 : x 2 : x 3 : x 4 ) P 3, and we readily check that this family, while isotrivial, is not constant and thus yields infinitely many distinct specializations to curves of genus g with at least the y 2 = Ax m + B has genus g and at least the 8m = 16(g + 1) rational points (ζx i, ±Q(x i )) (with ζ m = 1) over K. The Brumer-Mestre construction requires a large subgroup G of PGL 2 (K), and so cannot work over Q since the largest subgroups of PGL 2 (Q) have 4

size 12. To get hyperelliptic curves of genus g with many rational points over Q, we could apply Lemma M with r = 2 and d = 2g + 3 to write the polynomial P (x) = 4g+6 i=1 (x x i) (x i Q distinct) as Q 2 R with R usually of degree 2g + 2; then the curve y 2 = R(x) has genus g and at least the 8g + 12 rational points (x i, ±Q(x i )). 3. Slicing surfaces with many lines Let S be a (smooth 2 ) surface in P r (usually P 3 ) with N lines. Then a generic (hyper)plane slice of S comes with N rational points. For instance if r = 3 and S has degree d then N(g) N where g = (d 1)(d 2)/2. But the geometric question of maximizing the number of lines on a smooth surface of degree d is still open. Note: It s natural to look at sliced surfaces because that s what a parametrized family of curves is! The Fermat surface Z d + Y d + Z d + T d = 0 has 3d 2 lines over C; this is the largest known for all d > 2 except 4, 6, 8, 12, 20. For d = 4, Schur s surface S : X 4 +XY 3 = Z 4 +ZT 3 has 64 lines [Sc, p.291], and this is best possible ([Se], which also gets a bound (d 2)(11d 6) < 11d 2 bound for all d > 2). Note: In general if P is a homogeneous polynomial of degree d whose configuration of zeros in P 1 has M symmetries then the surface P (X, Y ) = P (Z, T ) has Md + d 2 lines. For d 4, 6, 8, 12, 20 the largest M occurs for polynomials such as X d + Y d yielding the Fermat surface. But the vertices of a regular polyhedron (a.k.a. Platonic solid ) inscribed in the Riemann sphere improve on this for the above five exceptional d. For instance Schur s quartic arises from a regular tetrahedron. Of course surfaces not of the form P (X, Y ) = P (Z, T ) might conceivably have more lines once d > 4. So N(3) 64. In fact the pencil through a random line l 0 has 4 basepoints, for a total of 68, and further finagling 3 raises this to 72. 2 Mild singularities in S may be tolerated, but we can t allow e.g. a cone over a curve. 3 Let l, l S be the lines (x : y : x : y) and ( 3x : 3y : x + y : 2x y). These meet at a point, and thus span a plane, namely (y t) = ( 3 + 1)(z x). The intersection of this plane with S is the union of l, l and a conic c 1. Let c 2 be another such conic, in a different plane. The planes through l 0 whose points of intersection with c i are rational for i = 1, 2 are then parametrized by an elliptic curve. Over a large enough number field that curve has infinitely many points, producing infinitely many plane quartics with 68 + 2 2 = 72 rational points. 5

Also: pick a rational curve C on S of degree > 1. Then the slice of S tangent to a generic point of C is quartic with node, so of genus 2, with 64 points. The hyperelliptic involution gives 64 more, for a total of 128, the current record. Likewise, slicing the sextic surface S : XY (X 4 Y 4 ) = ZT (Z 4 T 4 ), which has 180 lines and an elliptic curve 4 E, we obtain N(9) 180 (the current record) and N(10) 187. Each line l S is disjoint from 145 others, so the pencil of sections through l consists of quintic plane curves with at least 145 points. Using E again we find N(6) 145 + 1 = 146, currently best known. (A degree-12 sextic with 864 lines, each disjoint from 781 others, gives N(45) 781 in the same way.) 4 There are planes containing three of the 180 lines; the intersection of such a plane with S thus consists of those lines and a residual curve of degree 3, which turns out to be nonsingular. This can be seen by using P (X, Y ) = (X 3 Y 3 )(8X 3 + Y 3 ) instead of P (X, Y ) = XY (X 4 Y 4 ), in effect rotating the Riemann sphere to display the threefold symmetry of the octahedron rather than the fourfold one. The plane t = y cuts this surface in 3 lines and the elliptic curve X 3 + 7Y 3 + Z 3 = 0, which happens to even have infinitely many rational points over Q. 6

3 & 3. Highly symmetric curves Harris observed: Brumer and Mestre s families are both pencils of curves with 4g + 4 symmetries. But a 1-parameter family of curves of genus g may have as many as 12(g 1) automorphisms (Riemann-Hurwitz) so may do better than 16(g + 1), especially if the family is again a linear pencil. [NB Brumer s record curves over Q of genus 2,3 both fall on such pencils. 5 ] For instance the pencil of genus-4 curves C t : y 3 + 1 y 3 1 = t x3 + 1 x 3 1 has 36 automorphisms and 18 base points. For certain t parametrized by an elliptic curve there are at least 3 orbits of 36 rational points, so N(4) 18 + 3 36 = 126. These rational points require adjoining at least the cube roots of unity, but a twisted version of the C t works over Q: C t : y 3 9y y 2 1 = t x3 9x x 2 1 Fortunately the relevant elliptic curve 6 Y 2 + Y = X 3 480X + 230400 (conductor 4830) has positive rank over Q; thus B(4, Q) 126. Example: C t with t = 9717/2288. This is the only g for which the records for B(g, Q), N(g, Q) and N(g) are equal. Similarly the pencil of plane quartics X 4 + Y 4 + Z 4 = t(x 2 Y 2 + X 2 Z 2 + Y 2 + Z 2 ) with 24 automorphisms and 8 base points yields 7 N(3) 8 + 3 24 = 80 (but alas not over Q). 5... as do the new Kulesz and Stahlke curves of genus 2. 6 This is the curve parametrizing t such that the three roots of f(4z 3)/f(z) = t are rational. 7 Restricting (X : Y : Z) to a generic line through one of the base points such as (1 : e 2πi/3 : e 4πi/3 ) makes t a function of degree 3, so again the t s whose three preimages are all rational are parametrized by an elliptic curve, which will have infinitely many points over some number field. 7

Note: Numerical computation reveals 20 more C t with 3 orbits not accounted for by the elliptic parametrization (e.g. t = 2144/187), but none with 4. Variation ( construction 3 ): isotrivial pencils like Mestre s y 2 = Ax m +B. In genus 3, C A : AZ 4 = X 4 XY 3 has 48 symmetries and 4 base points. It s easy to get one orbit; for two use X : Y = x 4 : y 4 and y 4 : x 4, finding the record N(3) 4 + 2 48 = 100. Two more records come from homogeneous P (X, Y ) of degree 12 with icosahedral symmetry: N(5) 12 + 120 = 132 using AZ 2 = P (X, Y ), and N(10) 12 + 180 = 192 from AZ 3 = P (X, Y ). Record curves over Q. For large g, Mestre s 8g +C is the best lower bound known for both B(g, Q) and N(g, Q). However, small g, in particular g = 2, 3, much effort has been spent As of May 1996, the best lower bounds known for B(g, Q) for g = 2, 3, 4, 5 come from the following record curves: genus 2. In [KK] Kulesz and Keller give the curve C 2 : y 2 = 278271081x 2 (x 2 9) 2 229833600(x 2 1) 2 (9) of genus 2 with 12 automorphisms (coming from the S 3 automorphisms of the projective x-line preserving {0, 1, }) and at least 12 49 = 588 points rational over Q, with x-coordinates in the 49 orbits represented by 10, 13/3, 25/7, 29/8, 35/9, 37/7, 41/11, 50/7, 52/3, 85/4, 97/3, 100/19, 109, 119/33, 125/11, 142/17, 173/14, 197/25, 239/13, 364/95, 365/101, 911/159, 1079/307, 1361/195, 1388/321, 1858/539, 2020/573, 2324/379, 2359/247, 2626/745, 4913/601, 5146/1493, 5371/1409, 5899/1617, 6017/1511, 6599/1893, 8995/1812, 13675/3913, 14812/2557, 22388/2545, 52718/14131, 96325/22009, 172294/48663, 175708/17645, 234127/63521, 329183/58789, 344923/54182, 427775/94857, 4207979/680553. The automorphism group acts irreducibly on the space of holomorphic differentials on this curve; thus (for instance using the criterion of [ES]), the Jacobian J(C 2 ) is isogenous to the square of the elliptic curve E : y 2 = 278271081x(x 9) 2 229833600(x 1) 2 (10) which is the quotient of C 2 by the involution (x, y) ( x, y). The images of the known points on C 2 generate a subgroup of E of rank 12. Thus J(C 2 ) has rank at least 24. 8

genus 3. The record here is again due to Kulesz and Keller [KK]: The hyperelliptic curve C 3 : y 2 = 7920000(x 2 + 1) 4 136782591x 2 (x 2 1) 2 (11) has 16 automorphisms, coming from the 8-element dihedral group of automorphisms of the x-line preserving {0, ±1, }, and at least 16 11 = 176 points rational over Q, with x-coordinates in the 11 orbits represented by 3, 4, 5, 9, 25, 25/4, 59, 79/25, 97/30, 101/29, and 197/60. [Jacobian?] genus 4. The curves we used to show N(4) 126 are defined over Q, and thus also yield B(4, Q) 126. As noted already this is the best lower bound known. Conceivably some of the curves in this family might have one or more extra orbits of rational points; a few computer searches revealed no such examples but did find several other t values, such as 2144/187, References [CHM1] Caproaso, L., Harris, J., Mazur, B.: Uniformity of rational points. J. AMS, to appear. [CHM2] Caproaso, L., Harris, J., Mazur, B.: How many rational points can a curve have? Proc. of the Texel Conference, to appear. [ES] : Ekedahl, T., Serre, J.-P.: Exemples de courbes jacobienne complètement décomposable, C. R. Acad. Sci. Paris Sér. 1 317 (1993), 509 513. [Fa1] Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349 366. Erratum: Invent. Math. 75 (1984), 381. [Fa2] Faltings, G.: Diophantine approximations on abelian varieties, Ann. of Math. 133 (1991), 549 576. [KK] Kulesz, L., Keller W.: Courbes algébriques de genre 2 et 3 possédant de nombreux points rationnels. C. R. Acad. Sci. Paris Sér. 1 321 (1995) #11, 1469 1472. 9

[La] Lang, S.: Hyperbolic and diophantine analysis, Bull. AMS 14 #2 (1986), 159 205. [Sc] [Se] [St] Schur, F.: Ueber eine besondre Classe von Fläche vierter Ordnung. Math. Ann. 20 (1882), 254 296. Segre, B: The maximum number of lines lying on a quartic surface. Quart. J. Math. 14 (1943), 86 96. Stahlke, C.: Algebraic curves over Q with many rational points and minimal automorphism group. Preprint, 1996. 10