On the smallest point on a diagonal quartic threefold

Transcription

1 On the smallest oint on a diagonal quartic threefold Andreas-Stehan Elsenhans and Jörg Jahnel Abstract For the family x = a y +a 2 z +a 3 v + w,,, > 0, of diagonal quartic threefolds, we study the behaviour of the height of the smallest rational oint versus the Tamagawa tye number introduced by E Peyre Introduction A comuter exeriment Let V P n É be a Fano variety defined over É If V É ν for every ν ValÉ then it is natural to ask whether V É Hasse s rincile Further, it would be desirable to have an a-riori uer bound for the height of the smallest É-rational oint on V as this would allow to effectively decide whether V É or not When V is a conic, Legendre s theorem on zeroes of ternary quadratic forms roves the Hasse rincile and, moreover, yields an effective bound for the smallest oint For quadrics of arbitrary dimension, the same is true by an observation due to J W S Cassels [Ca] Further, there is a theorem of C L Siegel [Si, Satz ] which rovides a generalization to hyersurfaces defined by norm equations For more general Fano varieties, no theoretical uer bound is known for the height of the smallest É-rational oint Some of these varieties fail the Hasse rincile In this note, we resent some theoretical and exerimental results concerning the height of the smallest É-rational oint on quartic hyersurfaces in P É 2 There is a conjecture, due to Yu I Manin, that the number of É-rational oints of anticanonical height < B on a Fano variety V is asymtotically equal to τb log rkpicv B, for B Key words and hrases Diagonal quartic threefold, Diohantine equation, smallest solution, naive height, E Peyre s Tamagawa-tye number The comuter art of this work was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientific Comuting at the Göttingen Mathematical Institute Both authors are grateful to Prof Y Tschinkel for the ermission to use these machines as well as to the system administrators for their suort

2 In the articular case of a quartic threefold, the anticanonical height is the same as the naive height Further, rk PicV = and, finally, the coefficient τ Ê equals the Tamagawa-tye number τv introduced by E Peyre in [Pe] Thus, one exects τv B oints of height <B Assuming equidistribution, the height of the smallest oint should be Being a bit otimistic, this might lead to the exectation τv that mv, the height of the smallest É-rational oint on V, is always less than C τv for a certain absolute constant C 3 To test this exectation, we comuted the Tamagawa number and ascertained the smallest É-rational oint for each of the quartic threefolds V a,b P É given by ax = by + z + v + w for a, b =,, 000 On of these varieties, there are no É-rational oints as the equation is unsolvable in É for = 2, 5, or 29 Note that, for each rime different from 2, 5, or 29, there are -adic oints already on the Fermat quartic given by z +v +w = 0 On each of the remaining varieties, É-rational oints were found In other words, there are no counterexamles to the Hasse rincile in this family The methods to systematically search for solutions of Diohantine equations and to comute Tamagawa numbers we alied here are described in our earlier aers [EJ, EJ2] The results are summarized by the diagrams below a, b = 00 a, b = 000 Figure : Height of smallest oint versus Tamagawa number It is aarent from the diagrams that the exeriment agrees with the exectation above The sloe of a line tangent to the to right of each of the scatter lots is indeed near However, we show in Section 2 that, in general, the inequality mv < C does not hold The following remains a logical ossibility τv 2

3 Question For every ε > 0, does there exist a constant Cε such that, for each quartic threefold, mv < Cε τv? +ε 5 Peyre s constant In our situation, rk PicV É = In articular, there is no Brauer-Manin obstruction on V Recall that, in this case, E Peyre s Tamagawa-tye number is defined [PT, Definition 2] as an infinite roduct τv := ν ValÉ τ νv For a rime number, the definition of the local factor may be simlified to τ V := #V / n lim n 3n τ V is described in [Pe, Lemme 57] In the case of the diagonal quartic threefold V,, given by x x = 0 < 0, a,, > 0 in P É, this yields τ V,, = dy dz dv dw a y + a 2 z + a 3 v + w 3/ where the domain of integration is R R := {y, z, v, w [, ] a y + a 2 z + a 3 v + w } 6 In the case of diagonal quartic threefolds, there is an estimate for mv in terms of τv Namely, admits a fundamental finiteness roerty More recisely, in Section 3, we will show the following τv results Theorem Let a =,, be a vector such that,,, < 0, and a,, > 0 Denote by V a the quartic in P given by a É 0x x = 0 Then, for each ε > 0 there exists a constant Cε > 0 such that τv a Cε H naive ε Corollary Fundamental Finiteness For each B > 0, there are only finitely many quartics V a : x x = 0 in P É such that < 0, a,, > 0, and τv a > B Corollary An inefficient search bound There exists a monotonically decreasing function S : 0, [0,, the search bound, satisfying the following condition Every quartic V a : x x = 0 such that < 0 and a,, > 0 admits, if V a É, already a É-rational oint of height SτV a Proof One may simly ut St := max τv a t V a É min P V a É hp In other words, we have mv a SτV a as soon as V a É 3

4 2 A negative result For a Æ, let V a P É be given by ax = y + z + v + w and let mv a := min { H naive x : y : z : v : w x : y : z : v : w V a É} be the smallest height of a É-rational oint on V a We comare mv a with the Tamagawa tye number τ a := τv a 2 Notation For a rime number and integers y, z,, not all of which are equal to zero, we write gcd y, z, for the largest ower of dividing all of the y, z, 22 Theorem There is no constant C such that mv a < for all a Æ Proof The roof consists of several stes C τ a First ste For a, one has τ a = a I where I is an integral indeendent of a This follows immediately from formula above Second ste For the height of the smallest oint, we have mv a a x yields y + z + v + w a and max{ y, z, v, w } a Third ste There are two ositive constants C and C 2 such that, for all a Æ, C < τ a < C 2 >3, a For a rime of good reduction, Hensel s lemma shows τ V a = #V a 3 Further, for the number of oints on a non-singular variety over a finite field, there are excellent estimates rovided by the Weil conjectures, roven by P Deligne In our situation, [De, Théorème 8] may be directly alied It shows #V a = E a with an error-term E a 60 3/2 Note that dim H 3 V, Ê = 60 for every smooth quartic threefold V in P Consequently, 60 / + 60 / 3/2 side is ositive for > 3 resectively + 60 / is convergent 3/2 τ a Here, the left hand 3/2 The infinite roduct over all 60 / 3/2 Fourth ste There is a sequence {a i } i Æ of natural numbers such that { } τ 3 or a i a i i Æ

5 is unbounded Let C Ê be given We will show that 3 or a τ a > C when a := r is a roduct of sufficiently many different rimes i 3 mod fulfilling a mod M for M := Let be a rime such that a We made sure that a Then, for any oint x : y : z : v : w V a / n, the assumtion y, z, v, w would imly ax and x Therefore, gcd y, z, v, w = Further, y + z + v + w 0 mod As 3 mod, the number of solutions of that congruence is the same as that of y 2 +z 2 +v 2 +w 2 0 mod Since remains rime in [i], this quadratic form is a direct sum of two norm forms Its number of zeroes in is therefore equal to gcd y, z, v, w = imlies that Hensel s lemma is alicable It shows #V a / n = n + 2 3n n = 3n Hence, τ a = = + We may consequently write r τ a = + τ i 2 i 3 a i 3 or a i= 3 The second roduct is over = 2 and a finite number of rimes of good reduction The value of the roduct deends only on the residue of a modulo M = by virtue of [EJ2, Lemma 3a and bii] In articular, our assumtion a mod M imlies T := 3 τ a is a constant It is clear that T > 0 as the equation x = y + z + v + w admits a non-zero solution in /m for any m > It remains to show that there exists a set {,, r } of rimes i 3 mod such that i > 3, + r i= and r mod M + i 2 i 5 C 3 i T,

6 Condition + is easily satisfied as the series 3 mod diverges We find a set {,, s } of rime numbers i > 3 of the form i 3 mod such that s i= + i 2 i 3 i C T Enlarging {,, s } makes that roduct even bigger We may therefore assume s 3 mod The numbers M and s are relatively rime By Dirichlet s rime number theorem, there exists a rime s+, larger than each of the,, s, such that s+ mod M This shows, in articular, s+ 3 mod The assertion follows Conclusion The four stes together show that mv a τ a is unbounded 3 The fundamental finiteness roerty 3 An estimate for the factors at the finite laces 3 Notation i For a rime number and an integer x 0, we ut x := νx Note x = / x for the normalized -adic valuation ii By utting νx := min νξ, we carry the -adic valuation from over ξ to / r x=ξ mod r Note that any 0 x / r has the form x = ε νx where ε / r is a unit Clearly, ε is unique only in the case νx = 0 32 Definition For,, 5, r Æ, and ν 0,,ν r, ut S r ν 0,,ν ;,, := {x 0,,x / r 5 νx 0 = ν 0,,νx = ν ; x x = 0 / r } For the articular case ν 0 = = ν = 0, we will write Z r,, := S r 0,,0;,, Ie, Z r,, = {x 0,,x [ / r ] 5 x x = 0 /r } We will use the notation z r,, := #Z r,, 33 Sublemma If k,, and r > k then we have z r,, = 5k z r k / k,, / k Proof Since x x = k / k x / k x, there is a surjection ι: Z r,, Z r k / k,, / k, given by x 0,,x x 0 mod r k,,x mod r k The kernel of the homomorhism of modules underlying ι is r k / r 5 6

7 3 Lemma Assume gcd,, = k Then, there is an estimate z r,, 8 r+k Proof Suose first that k = 0 This means, one of the coefficients is rime to Without restriction, assume For any x,,x / r, there aears an equation of the form x 0 = c For odd, it cannot have more than four solutions in / r as this grou is cyclic On the other hand, in the case = 2, we have /2 r = /2 r 2 /2 and u to eight solutions are ossible The general case follows directly from Sublemma 33 Indeed, if k < r then z r,, = 5k z r k / k,, / k 5k 8 r k = 8 r+k On the other hand, if k r then the assertion is comletely trivial since z r,, = #Z r,, < 5r r+k < 8 r+k 35 Remark The roof shows that in the case 2 the same inequality is true with coefficient instead of 8 If 3 mod then one could even reduce the coefficient to 2 Unfortunately, these observations do not lead to a substantial imrovement of our final result 36 Lemma Let r Æ and ν 0,,ν r Then, #S ν r 0,,ν ;,, = zr ν 0,, ν ϕ r ν 0 ϕ r ν ϕ r 5 Proof As ν 0 x ν x = ν 0 x ν x, we have a surjection π: Z r ν 0,, ν S ν r 0,,ν ;,,, given by x 0,,x ν 0 x 0,, ν x For i = 0,,, consider the maing ι: / r / r, x 0 ν i x 0 If ν i = r then ι is the zero ma All ϕ r = r units are maed to zero Otherwise, observe that ι is ν i : on its image Further, νιx = ν i if and only if x is a unit By consequence, π is K ν 0 K ν : when we ut K ν := ν for ν r and K r := r Summarizing, we could have written K ν := ϕ r /ϕ r ν The assertion follows 37 Corollary Let,, \{0} 5 Then, for the local factor := τ V,,, one has τ,, τ,, = lim r r ν 0,,ν =0 z r ν 0,, ν ϕ r ν 0 ϕ r ν r ϕ r 5 7

8 Proof [PT, Corollary 35] imlies that τ,, = lim r r ν 0,,ν =0 #S r ν 0,,ν ;,, r Lemma 36 yields the assertion 38 Proosition Let,, \ {0} 5 Then, for each ε such that 0 < ε <, one has τ,, 8 Proof By Lemma 3, Writing k i := a i, we see ε a 0 a ε 3 ε z r ν 0,, ν / r 8 gcd ν 0,, ν = 8 gcd ν 0 a 0,, ν a z r ν 0,, ν / r 8 gcd ν 0+k 0,, ν +k = 8 min{ν 0+k 0,,ν +k } a We estimate the minimum by a weighted arithmetic mean with weights ε ε, and ε,, ε min{ν 0 + k 0,, ν + k } ε ν 0 + k ε ε ν 3 + k 3, ε, + εν + k This shows = εν ν 3 + εν + ε k k 3 + εk z r ν 0,, ν / r 8 εν 0++ν 3 +εν + ε k 0++k 3 +εk = 8 εν 0++ν 3 a +εν 0 a ε 3 a ε We may therefore write τ,, 8 a 0 a 3 r lim r ν 0,,ν =0 ε a ε εν 0++ν 3 +εν ϕ r ν 0 ϕ r ν ϕ r 5 8

9 Here, the term under the limit is recisely the roduct of four coies of the finite sum r ν=0 and one coy of the finite sum εν ϕ r ν ϕ r = r ν=0 ε + ν ε r r ν=0 εν ϕ r ν ϕ r = r ν=0 ε + ν ε r For r, geometric series do aear while the additional summands tend to zero 39 Remark Unfortunately, the constants C ε := 8 ε ε have the roerty that the roduct Cε diverges On the other hand, we have at least that C ε is bounded for, say C ε C ε 30 Lemma Let C > be any constant Then, for each ε > 0, one has C c x ε x for a suitable constant c deending on ε Proof This follows directly from [Na, Theorem 72] together with [Na, Section 7, Exercise 7] 3 Proosition For each ε > 0, there exists a constant c such that τ,, c min a i ε i=0,, rime for all,, \{0} 5 Proof As already noticed in the third ste of the roof of Theorem 22, the roduct over all rimes of good reduction is bounded by consequence of the Weil conjectures It, therefore, remains to show that,, c min a i ε i=0,, τ rime 2 For this, we may assume that ε is small, say ε < 9

10 Then, by Proosition 38, we have at first τ,, C ε = C ε a 0 a ε5 3 a 5 ε a 0 a 3 a ε5 a +ε Here, the indices 0,, are interchangeable Hence, it is even allowed to write τ,, a C ε 0 a ε5 max a +ε i = C ε a 0 a i ε5 min i a i ε Now, we multily over all rime divisors of 2 Thereby, on the right hand side, we may twice write the roduct over all rimes since the two rightmost factors are equal to one for, anyway,, ε a ε5 min a i ε i=0,, τ rime 2 C rime 2 ε C rime 2 = ε 5 min a i ε i=0,, when we observe that a = a Lemma 30, Further, we have C ε C ε c 2 2 We finally estimate 2 ε 5 by a constant The assertion follows ε 5 C ε and, by 32 A bound for the factor at the infinite lace := τ V,, described in Para- We want to estimate the integral τ,, grah 5 32 Lemma There exist two constants C and C 2 such that C min, 2 /, if, τ,, a0 a C min, 2 / + C 2 a / a0 a log min,3a, otherwise, for all,, Ê 5 satisfying a a 2 a 3 and Proof A linear substitution shows τ,, = dy dz dv dw a0 a y + z + v + w 3/ R 0 0

11 where R 0 := {y, z, v, w [ a /, a / ] [ a /, a / ] y + z + v + w } The integrand is non-negative We cover R 0 R R 2 by two sets as follows, R := {y, z, v, w Ê y + z + v + w min, 2 }, R 2 := {y, z, v, w Ê y + z + v min, 3a and w [ a /, a / ]}, and estimate In the case, the domain of integration is covered by R alone and we may omit R 2, comletely By homogeneity, we have R y + z + v + w 3/ dy dz dv dw = ω min,2 / 0 r 3 r3 dr = ω min, 2 / where ω is the three-dimensional hyersurface measure of the l -unit hyershere Further, R 2 where S := {y, z, v, w Ê y + z + v + w = } dy dz dv dw y + z + v + w 2a/ 3/ R 3 dy dz dv y + z + v 3/ R 3 := {y, z, v Ê 3 y + z + v min, 3a } The latter integral may be treated in much the same way as the one above We see R 3 y + z + v 3/ dy dz dv = ω 2 min,3a / a / r 3 r2 dr where ω 2 is the usual two-dimensional hyersurface measure of the l -unit shere Finally, min,3a / a / S 2 := {y, z, v Ê 3 y + z + v = } r 3 r2 dr = log min, 3a / a / = log min, 3a

12 322 Proosition For every ε > 0, there exists a constant C such that τ,, C +ε min i=0,, a i for each,, 5 satisfying < 0 and a,, > 0 Proof We assume without restriction that a There are two cases to be distinguished First case Then, by Lemma 32, we have τ,, C min{, 2 } = C a0 = C min i=0,, a i Second case > Here, Lemma 32 shows τ,, = a C min{, 2 } + C2 a / C 2 + C2 a / log C 2 + C2 log log min{, 3a } = min a i C 2 + C2 log a 0 i=0,, min i=0,, a i C 2 + C2 log min i=0,, a i C 3 ε 33 The Tamagawa number 33 Proosition For every ε > 0, there exists a constant C > 0 such that τ C Hnaive,, ε for each,, 5 satisfying < 0 and a,, > 0 Proof By Proosition 322, we have τ,, C + ε 2 min i=0,, a i 2

13 On the other hand, by Theorem 3, It follows that rime τ,, τ,, C 3 ε 2 and C 2 τ,, C 3 It is obvious that max This shows ε 2 min a i i=0,, min a i min a i i=0,, i=0,, [ rime [ ] [ ] min i min a i rime i=0,, i=0,, [ ε 2 min a ] ε 2 i i=0,, = max rime i=0,, a i max i=0,, a i C 3 [ ] ε ε 2 2 = H naive a 0 C 3 [ ε 2 i=0,, a i τ,, C 3 a 0 a and max i=0,, a i max i=0,, a i a ] ε 2 H naive ε 2 ε 2 ] min a i i=0,, ε 2 0 a = = Hnaive C 3 ε 332 Lemma Let P É be any oint such that 0,, 0 Then, H naive H naive Proof First, observe that is a welldefined ma Hence, we may assume without restriction that,, and gcd,, = This yields H naive = max a i i=0,, 3

14 On the other hand, = a a 2 a 3 a a 2 a 3 Consequently, H naive [ max i=0,, a i ] = H naive are inter- From this, the asserted inequality emerges when the roles of a i and a i changed 333 Corollary Let,, such that gcd,, = Then, H naive 20 Proof Observe that max a i 5 = H naive 5 and aly i=0,, Lemma Theorem For each ε > 0, there exists a constant Cε > 0 such that, for all,, 5 satisfying < 0 and a,, > 0, τ Cε H,, naive ε Proof We may assume that gcd,, = Then, by Proosition 33, τ C Hnaive,, ε 20 Corollary 333 yields ε 20 H naive ε 335 Corollary Fundamental finiteness For each B > 0, there are only finitely many quartics V,, : x x = 0 in P É such that < 0, a,, > 0, and τ,, > B Proof This is an immediate consequence of the comarison to the naive height established in Theorem 33 References [Ca] Cassels, J W S: Bounds for the least solutions of homogeneous quadratic equations, Proc Cambridge Philos Soc 5 955, [De] Deligne, P: La conjecture de Weil I, Publ Math IHES 3 97, [EJ] Elsenhans, A-S and Jahnel, J: The Diohantine equation x + 2y = z + w, Math Com ,

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