Genus bounds for curves with fixed Frobenius eigenvalues
aa r X i v : . [ m a t h . N T ] J a n GENUS BOUNDS FOR CURVESWITH FIXED FROBENIUS EIGENVALUES
NOAM D. ELKIES, EVERETT W. HOWE, AND CHRISTOPHE RITZENTHALER
Abstract.
For every finite collection C of abelian varieties over F q , we pro-duce an explicit upper bound on the genus of curves over F q whose Jacobiansare isogenous to a product of powers of elements of C .Our explicit bound is expressed in terms of the Frobenius angles of theelements of C . In general, suppose that S is a finite collection of s real numbersin the interval [0 , π ]. If S = { } set r = 1 /
2; otherwise, let r = S ∩ { π } ) + 2 X θ ∈ S \{ ,π } l π θ m . We show that if C is a curve over F q whose genus is greater thanmin (cid:18) s q s log q, ( √ q + 1) r (cid:18) q − r (cid:19)(cid:19) , then C has a Frobenius angle θ such that neither θ nor − θ lies in S .We do not claim that this genus bound is best possible. For any particularset S we can usually obtain a better bound by solving a linear programmingproblem. For example, we use linear programming to give a new proof of aresult of Duursma and Enjalbert: If the Jacobian of a curve C over F isisogenous to a product of elliptic curves over F then the genus of C is atmost 26. As Duursma and Enjalbert note, this bound is sharp, because thereis an F -rational model of the genus-26 modular curve X (11) whose Jacobiansplits completely into elliptic curves.As an application of our results, we reprove (and correct a small error in)a result of Yamauchi, which provides the complete list of positive integers N such that the modular Jacobian J ( N ) is isogenous over Q to a product ofelliptic curves. Introduction
Let ( C n ) be a sequence of curves over a finite field k such that the genus of C n tends to infinity with n . Serre [15, Cor. 2, p. 93] applies a result of Tsfasman andVl˘adut¸ [18, 19] to show that the dimension of the largest k -simple isogeny factorof the Jacobian of C n tends to infinity with n . In this article we give an explicitbound for this asymptotic result.The Weil polynomial of a d -dimensional abelian variety A over a finite field F q is the characteristic polynomial of the q -th power Frobenius endomorphism of A (acting, for instance, on the ℓ -adic Tate module of A ). The Weil polynomial is a Date : 14 December 2011.2010
Mathematics Subject Classification.
Primary 14G10; Secondary 11G20, 14G15, 14H25.
Key words and phrases.
Curve, Jacobian, Weil polynomial, Frobenius eigenvalue, genus, linearprogramming.The third author is partially supported by grant MTM2006-11391 from the Spanish MEC andby grant ANR-09-BLAN-0020-01 from the French ANR. monic polynomial in Z [ x ] of degree 2 d , and its complex roots α , . . . , α d all havemagnitude √ q . The roots can be written α j = √ q exp( iθ j ) for real numbers θ j inthe half-open interval ( − π, π ], and the roots can be ordered so that α j α j + d = q and 0 ≤ θ ≤ θ ≤ · · · ≤ θ d ≤ π. The θ j are called the Frobenius angles of A , and the θ j that are nonnegative are the nonnegative Frobenius angles of A . If C is a curve over F q , we define the Frobeniusangles of C to be the Frobenius angles of its Jacobian.We show that for any finite set S of real numbers in the interval [0 , π ], everycurve over F q of sufficiently large genus has a nonnegative Frobenius angle thatdoes not lie in S . Theorem 1.1.
Let S be a finite set of s real numbers θ with ≤ θ ≤ π . If S = { } set r = 1 / ; otherwise, take r = S ∩ { π } ) + 2 X θ ∈ S \{ ,π } l π θ m , where ⌈ x ⌉ denotes the least integer greater than or equal to the real number x . Let B = 23 s q s log q and B = ( √ q + 1) r (cid:18) q − r (cid:19) . If C is a curve over F q whose nonnegative Frobenius angles all lie in S , then thegenus g of C satisfies g ≤ B and g < B .Remark . Elementary calculations show that if s > √ q log q then the bound B from Theorem 1.1 is smaller than the bound B .Theorem 1.1 allows us to derive an explicit version of Serre’s result. Corollary 1.3. If C is a curve of genus g > over a finite field F q , then theJacobian of C has a simple isogeny factor of dimension greater than s log log g q . The first genus bound from Theorem 1.1 leads to a corollary that does notmention Frobenius angles.
Corollary 1.4.
Let A be a d -dimensional abelian variety over F q . If C is a curveover F q of genus greater than d q d log q then the Jacobian of C has a simpleisogeny factor B that is not an isogeny factor of A . One natural choice for the set S in Theorem 1.1 is the set of all nonnegativeFrobenius angles for elliptic curves over a given finite field. Applying the theoremto this set leads to the following corollary: Corollary 1.5.
Suppose C is a curve over F q whose Jacobian is isogenous over F q to a product of elliptic curves. Then the genus of C is at most q √ q +3 log q . We prove Theorem 1.1 and its corollaries in Section 2. As we will show inRemark 2.6, for every prime power q there is a family of sets S for which (inthe notation of Theorem 1.1) the ratio B /g is less than 47 s log q ; similarly, inRemark 2.4 we show that there is a family of examples with s = 1 and q → ∞ for which the ratio B /g approaches 1. For many sets S , however, we expect that ENUS BOUNDS FOR CURVES 3 the bounds given by Theorem 1.1 are likely to be far from optimal; in Section 3we show how tighter bounds can sometimes be obtained by solving an integerlinear programming problem. As a concrete example, we take S to be the setof nonnegative Frobenius angles of the elliptic curves over F , and use the linearprogramming method to show that the genus of a curve over F whose Jacobiansplits up to isogeny into elliptic curves is at most 26, a result proved earlier byDuursma and Enjalbert [8] by a different (but related) method. In Section 4 weshow that for this particular set S the genus bound of 26 is sharp, because thegenus-26 modular curve X (11) has a model over F whose Jacobian is isogenousover F to a product of elliptic curves. (We find sharp upper bounds in other casesas well; see Remark 3.7.) In Section 5 we use this genus bound to give a simpleproof of a result of Yamauchi [21] on the values of N for which the Jacobian of themodular curve X ( N ) is isogenous over Q to a product of elliptic curves. Conventions and notation.
Operators such as Hom, End or Aut applied to varietiesover a field k will always refer to k -rational homomorphisms and endomorphisms.Similarly, when we say that an abelian variety over k is ‘decomposable’, ‘split’, or‘simple’, these words should be interpreted with respect to isogenies over k . In thesequel, we use e ( x ) to denote exp( ix ), where i = √−
1, and we use Re( x ) to denotethe real part of a complex number x . Acknowledgments.
After an initial version of this work was posted on the arXiv,Mike Zieve alerted us to the paper of Duursma and Enjalbert [8] mentioned above,which contains a result that implies our Lemma 2.1 (see § F whose Jacobian is isogenous to aproduct of elliptic curves has genus at most 26. The print version of their paperalso states that there is a model of X (11) over F whose Jacobian is isogenous over F to a produce of elliptic curves; this is proved in an addendum added to thearXiv version. We are grateful to Mike Zieve for telling us about [8], and to IwanDuursma for discussions about these results.We are grateful to Armand Brumer for informing us of Yamauchi’s paper [21].In the initial version of this paper, the bound B in Theorem 1.1 was a doubly-exponential expression in s that we obtained via an argument using our Lemma 2.1.We are extremely grateful to Zeev Rudnick and Sergei Konyagin for pointing outto us that the work of Smyth [16] could be used to get a bound that is singly-exponential in s . Our proof that B gives an upper bound for the genus is duealmost entirely to them.The work in this paper was begun at the GeoCrypt 2009 conference in Pointe-`a-Pitre. We would like to thank the organizers of the conference for providing astimulating environment for collaboration.2. Proof of Theorem 1.1 and its corollaries
In this section we prove Theorem 1.1 and Corollaries 1.4 and 1.5. The argumentsthat show that B and B give upper bounds on the genus are independent of oneanother, so we break the proof of Theorem 1.1 into two parts. Proof of Theorem , part g ≤ B . The bound g ≤ B holds trivially when S is empty, so we may assume that s >
0. First we consider the case where s > C is a curve over F q all of whose Frobenius angles lie in S . Let theelements of S be θ , . . . , θ s , and for each index j let b j denote the multiplicity of θ j ELKIES, HOWE, AND RITZENTHALER as a Frobenius angle for C (unless θ j = 0 or θ j = π , in which case we let b j denotehalf of this multiplicity), so that we have b + . . . + b s = g. Reindex the θ j and the corresponding b j so that b ≥ b ≥ · · · ≥ b s , and note thatthis implies that b ≥ g/s . Finally, for every j let z j = e ( θ j ), and for every integer k > G ( k ) denote the weighted power sum G ( k ) = b z k + · · · + b s z ks . Weil’s ‘Riemann Hypothesis’ for curves over finite fields says that C ( F q k ) = q k + 1 − g X j =1 α kj = q k + 1 − q k/ Re G ( k ) , so we find that 2 Re G ( k ) = q k/ + q − k/ − C ( F q k ) /q k/ . Now we apply a result of Smyth [16]. Let β := ( b + · · · + b s ) /b , let λ be an arbitrary real number with 0 < λ ≤
1, and let K := 1 + ⌊ (4 β + 3) /λ ⌋ . Smyth shows that then ( b / − λ ) < max ≤ k ≤ K Re G ( k ) . In our case, we have β = g/b − ≤ s −
1, so K ≤ β + 3) /λ ≤ L , where weset L := 1 + (4 s − /λ. Also, for each k we have Re G ( k ) ≤ (1 / q k/ + q − k/ ). Since q x + q − x is anincreasing function of x for x >
0, Smyth’s result shows that( b / − λ ) < (1 / q L/ + q − L/ ) . As we noted earlier, we have b ≥ g/s , so for every λ with 0 < λ < gs < − λ (cid:16) q L/ + q − L/ (cid:17) . We will apply this inequality with λ = 1 − /a , where a = (cid:18) s − (cid:19) log q. (This choice of λ was obtained by taking the first two terms of the power seriesexpansion (in 1 /a ) of the value of λ that minimizes q L/ / (1 − λ ).) Note that L > s ≥
8, so that q L/ + q − L/ = q L/ (1 + q − L ) < q L/ . ENUS BOUNDS FOR CURVES 5
Combining this with (1), we find gs < (cid:18) (cid:19) (cid:18) − λ (cid:19) q L/ = (cid:18) (cid:19) (4 s − q ) exp(( L/
2) log q ) . Using the equality L = 1 + (4 s − /λ , we find that( L/
2) log q = (log q ) / (cid:18) s − (cid:19) log qλ = (log q ) / a/λ = (log q ) / a / ( a − q ) / a + a/ ( a − s log q + a/ ( a − . Since s ≥ q ≥ a ≥ (7 /
2) log 2 and exp( a/ ( a − < .
5. Thusexp(( L/
2) log q ) < . s log q ) = 5 . q s , so gs < (cid:18) (cid:19) (4 s )(log q )(5 . q s ) < sq s log q. This shows that g < B when s > s = 1. Let θ be the unique element of S . If θ ≥ π/ r defined in the statement of the theorem is equal to 2. It is easy to showthat then B < B , so the bound g < B , proved below, shows that g < B .The final case to consider is when s = 1 and θ < π/
2. If C has only onenonnegative Frobenius angle θ , then e ( θ ) √ q must be an algebraic integer of degreeat most 2 over the rationals, so the quantity t = e ( θ ) √ q + e ( − θ ) √ q is an integer.Furthermore, since θ is less than π/ t is positive. Then Weil’s theoremshows that 0 ≤ C ( F q ) = q + 1 − gt ≤ q + 1 − g, so that g ≤ q + 1 . Again, it follows easily that g < B . (cid:3) Our proof that B gives an upper bound for the genus relies upon the followinglemma. Lemma 2.1.
Let S be a set of real numbers in [0 , π ] and let T ∈ R [ X ] be apolynomial with nonnegative coefficients such that T (0) = 0 and Re( T ( e ( θ ))) ≥ for all θ ∈ S . If C is a curve over F q whose nonnegative Frobenius angles all liein S , then the genus of C is at most ( T ( q / ) + T ( q − / )) / .Proof. Let α , . . . , α g be the complex roots of the Weil polynomial of C , listedwith appropriate multiplicities, so that from Weil’s theorem we have C ( F q m ) = q m + 1 − g X j =1 α mj for all integers m >
0. Write T = a x + · · · + a n x n , a j ≥ . ELKIES, HOWE, AND RITZENTHALER
Then 0 ≤ n X m =1 a m C ( F q m ) q m/ = n X m =1 a m q m/ + q − m/ − g X j =1 α mj /q m/ = T ( q / ) + T ( q − / ) − g X j =1 T ( α j / √ q ) . By hypothesis, each summand T ( α j / √ q ) has real part at least 1, so0 ≤ T ( q / ) + T ( q − / ) − g, and hence g ≤ T ( q / ) + T ( q − / )2 . (cid:3) Remark . This lemma could also be proved by using the results in § Remark . Lemma 2.1 is stated for polynomials T with nonnegative coefficients,but an analogous statement holds when T is a power series in R [[ x ]] with nonneg-ative coefficients, so long as its radius of convergence exceeds √ q .With this lemma in hand, we complete the proof of Theorem 1.1. Our proof willdepend on a careful choice of the polynomial T . Proof of Theorem , part g < B . It is easy to check that g < B when S isempty, so we may assume that S is nonempty. If S = { } then the quantity r fromthe theorem is equal to 1 /
2. In this case we can take T = x , and we find that( T ( q / ) + T ( q − / )) / B . Then Lemma 2.1 shows that g < B . Solet us assume that S contains a nonzero element.Given a nonzero θ ∈ S , let m = (cid:6) π θ (cid:7) , so that cos( mθ ) ≤
0. If θ = π let P θ bethe polynomial 1 + x ; otherwise, set P θ = 1 − mθ ) x m + x m . In both caseswe have P θ ( e ( θ )) = 0 . Let P = Y θ ∈ S \{ } P θ . Then P is a polynomial with constant term 1, with nonnegative coefficients, andwith degree equal to r , where r = S ∩ { π } ) + 2 X θ ∈ S \{ ,π } l π θ m is as in the statement of the theorem. Let T = ( P − . Then T (0) = 0, thecoefficients of T are nonnegative, and for every θ ∈ S \ { } we have T ( e ( θ )) = 1 . Also, for each θ we have P θ (1) ≥
2, so P (1) ≥ T (1) ≥
1. Lemma 2.1 showsthat any curve whose nonnegative Frobenius angles all lie in S must have genus ENUS BOUNDS FOR CURVES 7 no larger than ( T ( q / ) + T ( q − / )) /
2. Now, for positive real numbers z we have1 ≤ P ( z ) ≤ (1 + z ) r , so that T ( z ) < (1 + z ) r . Thus we have T ( q / ) + T ( q − / ) < ( √ q + 1) r + (1 / √ q + 1) r = ( √ q + 1) r (1 + q − r ) , which gives the inequality g < B . (cid:3) Remark . In Section 3 we will see that our bounds can sometimes be bad.However, the following easy example shows that at least in one case our secondbound is asymptotically exact as q → ∞ . For any prime power q let E be asupersingular elliptic over F q with Weil polynomial x + q , corresponding to theset S = { π/ } . Applying Theorem 1.1, we see that the genus of a curve C/ F q withJacobian isogenous to a power of E is bounded above by( √ q + 1) · (cid:18) q − (cid:19) ∼ q / q → ∞ . On the other hand, Jac( C ) is isogenous to a power of E if and only if C is optimalover F q , in the sense that its number of points attains the Weil upper bound. Butthe maximal genus of such a curve is q ( q − / ∼ q / H q defined by x q +1 + y q +1 + z q +1 = 0 (see for instance [14]). Remark . The Hermitian curve H q , viewed as a curve over F q , again gives anexample of a curve whose genus comes close to the upper bound B . If we take S = { π } then the bound B for the field F q is( q + 1) · (cid:18) q − (cid:19) ∼ q / q → ∞ , while H q has genus q ( q − / Remark . Hermitian curves can also be used to give examples that limit theextent to which we might hope to improve the bound B . For any integer s > S be the s -element set S = (cid:26) π s , π s , . . . , (2 s − π s (cid:27) . Let q be any prime power and set Q = q s . Note that the nonnegative Frobeniusangles of a curve C over F q are contained in S if and only if the only nonnegativeFrobenius angle of the base extension of C to F Q is π/
2. Let C be the curve x Q +1 + y Q +1 + z Q +1 = 0 over F q . As noted in Remark 2.4, the base extensionof C to F Q has π/ C itself are contained in S . Since the genus of C is ( q s − q s ) /
2, we see thatTheorem 1.1 would be false if we replaced B with any expression of the form(polynomial in s and log q ) q s − ε for a positive constant ε .We end this section by proving the corollaries from the introduction. Proof of Corollary . We begin by noting that for every integer n >
0, thenumber of isogeny classes of n -dimensional abelian varieties over F q is less than6 n q n ( n +1) / . This is easy to check for n = 1 (see the proof of Corollary 1.5 below)and for n = 2, while for n > ELKIES, HOWE, AND RITZENTHALER
We leave the details of the argument to the reader, but we do at least note that itis helpful to observe that for every n , the quantity v n in [6, Thm. 1.2] is boundedabove by 264. It follows that the number of isogeny classes of simple abelian va-rieties over F q of dimension at most n is also less than 6 n q n ( n +1) / , and that thenumber of nonnegative Frobenius angles of abelian varieties over F q of dimensionat most n is less than n n q n ( n +1) / .Suppose, to obtain a contradiction, that the corollary is false, and let C be acurve of genus g over F q that provides a counterexample. Take n = $s log log g q % . For C to provide a counterexample we must have n ≥
1. We will apply Theo-rem 1.1 to the set of nonnegative Frobenius angles of abelian varieties over F q ofdimension at most n ; as we have just noted, this means that in Theorem 1.1 wehave s < n n q n ( n +1) / . Simple estimates for the terms in the right-hand side ofthis inequality show that(2) s < q n . Note that there are at least 5 isogeny classes of elliptic curves over any field, so s ≥
5; using this fact it is easy to show that the bound B = 23 s q s log q from thetheorem satisfies(3) B < q s . The definition of n tells us thatlog log g ≥ n log q, so exponentiating gives uslog g ≥ q n > q n log q > s log q, where the third inequality follows from (2). Exponentiating again, we find that g > q s > B , by (3). Theorem 1.1 then shows that Jac C has a simple isogeny factor of dimensiongreater than n , contradicting our assumption that C was a counterexample to thecorollary. This contradiction completes the proof. (cid:3) Proof of Corollary . Let S be the set of nonnegative Frobenius angles of theabelian variety A over F q , so that S ≤ d . If C is a curve over F q with genusgreater than 23 d q d log q then C has a nonnegative Frobenius angle θ not in S .Let B be any element of the unique isogeny class of simple abelian varieties over F q that have θ as one of their Frobenius angles. Then B is an isogeny factor of Jac C .On the other hand, A and B have coprime Weil polynomials, so by the Honda–Tatetheorem B is not an isogeny factor of A . (cid:3) Proof of Corollary . Let m be the largest integer with m ≤ q . The Weilpolynomial of an elliptic curve over F q is of the form x − tx + q , where t isan integer with − m ≤ t ≤ m . (Not every t in this range need come from anelliptic curve.) The nonnegative Frobenius angle θ t corresponding to a given t is θ t = cos − ( t/ (2 √ q )) . We take S to be the set { θ t : − m ≤ t ≤ m } . ENUS BOUNDS FOR CURVES 9
Clearly S ≤ √ q + 1. The bound B from Theorem 1.1 is then23 s q s log q ≤
23 (4 √ q + 1) q √ q +2 log q ≤ √ √ ! q √ q +3 log q< q √ q +3 log q. (cid:3) Linear programming
The arguments we used to prove Theorem 1.1 used no special properties of theset S and give a bound that is almost certainly far from optimal when applied tomost S . For example, suppose we take S to be the set of nonnegative Frobeniusangles coming from the elliptic curves over F . There are five isogeny classes ofelliptic curves over F (each containing a single curve), corresponding to the fivepossible traces of Frobenius − , − , , , t , let α t = ( − t + √ t − /
2, let θ t be the argument of α t , and let E t be an elliptic curve over F with trace t . Let us apply Theorem 1.1 to the set S = { θ t } . We find that B ≈ B ≈ . × . However, the best upperbound on the genus is much smaller than these numbers. Theorem 3.1 (Duursma and Enjalbert [8]) . Suppose C is a curve over F whoseJacobian is isogenous to a product of powers of the E t . Then the genus of C is atmost .Remark . Duursma and Enjalbert prove this result by using a stronger versionof our Lemma 2.1. We give the proof presented here as an example of our generaltechnique of using linear programming to get genus bounds.
Remark . We will see in Section 4 that the bound in Theorem 3.1 is sharp.
Proof of Theorem . Our bounds in Theorem 1.1 were obtained from the factthat the number of points on a curve over a finite field is always nonnegative. Weactually know a somewhat stronger constraint: For every n >
0, the number ofdegree- n places on a curve is nonnegative. If the size of the base field is largecompared to the genus of the curve in question, the bounds we get from usingplace counts are not much better than the one we get from point counts. However,Theorem 3.1 involves a very small field indeed.Let C be a curve as in the statement of the theorem. In particular, supposethere are integers e t ≥ C is isogenous to E e − − × E e − − × E e × E e × E e . Then for every n > C over F n is given by C ( F n ) = 2 n + 1 − X − ≤ t ≤ e t Tr( α nt ) , and the number N n of degree- n places on C is given by N n = 1 n X d | n µ ( n/d ) C ( F d ) , where µ is the M¨obius function. The nonnegativity of the number of degree- n places, for n = 1 , . . . ,
8, is expressedby the following inequalities:(4) − e − − e − + e +2 e ≤ e − − e − − e − e − e ≤ e − +2 e − − e − e ≤ − e − + e − +3 e + e − e ≤ e − − e − +2 e − e ≤ − e − + e − − e +3 e + e ≤ − e − +2 e − − e +2 e ≤ e − − e − +3 e − e +5 e ≤ P t = − e t ≤
26 for any nonnegative integers e t satisfying the inequal-ities above. Indeed we show this even if the e t are allowed to be nonnegative realnumbers. Maximizing their sum is then a linear programming problem. Solvingthe dual problem, we find that if we take 39 times the third inequality of (4), plus44 times the fourth inequality, plus 78 times the sixth inequality, plus 32 times theeighth inequality, we obtain72 e − + 72 e − + 72 e + 72 e + 72 e ≤ , so that e − + e − + e + e + e ≤ /
72 = 26, as claimed. (cid:3)
Remark . By enumerating 5-tuples of nonnegative integers ( e − , e − , e , e , e )that sum to 26 and checking whether they satisfy the inequalities (4), one findsthat if a genus-26 curve over F has a completely split Jacobian, then the Jacobianis isogenous to one of the varieties E − × E − × E × E × E ,E − × E − × E × E × E , or E − × E − × E × E × E . Remark . We would not get a genus bound if we looked only at the inequalitiescoming from counting the places of degree strictly less than 8. The reader can checkthat for any even g , the values e − = e = g/ e − = e = e = 0 satisfy thefirst seven inequalities in the system (4). On the other hand, a priori there is noreason to think that the bound we get from looking at the counts of places of degree8 and less will be the best possible; perhaps by using the nonnegativity of severalmore place counts, we would get a better bound. In this particular instance, resultsof Section 4 show that the bound we obtain is in fact sharp. Remark . A similar argument, using places of degree at most 12, shows thata curve over F whose Jacobian is isogenous to a product of elliptic curves musthave genus less than or equal to 2091. By extending the argument slightly (or byusing an integer linear programming package, such as the one in Magma [2]) onecan improve this upper bound to 2085. We suspect that this bound is not sharp! Remark . If one restricts to curves whose Jacobians are isogenous to a productof powers of ordinary elliptic curves, one finds a genus bound of 3 over F and 26over F . The first bound is reached by the the curve x + y + z + x y + x z + y z + x yz + xy z + xyz = 0which is a twist of the Klein quartic; the second bound is reached as well, as weshow in Section 4. ENUS BOUNDS FOR CURVES 11 The modular curve X (11)In this section we show that the modular curve X (11) has a model definedover F whose Jacobian is isogenous over F to a product of elliptic curves. Since X (11) has genus 26, this example shows that the bound of Theorem 3.1 is sharp.Duursma and Enjalbert [8] provide a different proof that X (11) has a model over F with completely split Jacobian; their argument, found in the addendum to thearXiv version of their paper, relies on working with Klein’s explicit model of X (11)in P .Let G be the twist of the finite group scheme Z / Z over F on which the F -Frobenius acts as multiplication by 3, and let G ′ be the Cartier dual of G , sothat G ′ is the twist of Z / Z on which the F -Frobenius acts as multiplicationby 2 / − e : G × G ′ → G m be the natural pairing from G × G ′ tothe multiplicative group.Let X be the modular curve over F whose non-cuspidal K -rational points, forevery extension K of F , parametrize pairs ( E, ϕ ), where E is an elliptic curveover K and ϕ is an isomorphism from the group scheme E [11] to the group scheme( G × G ′ ) ⊗ F K that takes the Weil pairing on E [11] to the pairing e . Theorem 4.1.
The genus of X is , and the Jacobian of X is isogenous to E − × E − × E × E × E , where each E t is an elliptic curve over F with trace t .Proof. The curve X is geometrically isomorphic to X (11), so it has genus 26and geometric automorphism group isomorphic to PSL (11) (see [13, Th´eor`eme 5]and [1]). Consider the group scheme ( G × G ′ ) ⊗ F F ; it is simply ( Z / Z ) , withthe F -Frobenius acting as multiplication by −
2. The automorphism group of this F -scheme is GL (11), and the subgroup of automorphisms that respect the pairing e is isomorphic to SL (11). There is a surjective mapAut (( G × G ′ ) ⊗ F F , e ) → Aut( X ⊗ F F )that sends an automorphism α of the finite group-scheme to the automorphism β of X ⊗ F F that takes a pair ( E, ϕ ) to (
E, αϕ ), and the kernel of this map isthe group {± } . Therefore all of the geometric automorphisms of X are alreadydefined over F .Using [10, Lemma 2.1] we see that the twists of the curve X ⊗ F F correspondto the conjugacy classes of Aut( X ⊗ F F ); also, the automorphism group of thetwist corresponding to the conjugacy class of an element α is isomorphic to thecommutator of α . Since PSL (11) has trivial center, we see that every nontrivialtwist of X ⊗ F F has automorphism group strictly smaller than PSL (11).Now take the Q ( √− Y of X (11) considered by Ligozat [12,Example 3.7.3, pp. 199–200]; Ligozat calls this curve X (11) K . Let p be theprime of Q ( √−
11) over 2, with residue field F . The automorphism group of Y is PSL (11), so the reduction of Y modulo p also has automorphism groupPSL (11). Therefore, the reduction of Y must be X ⊗ F F . Applying a resultof Ligozat [12, Prop 3.6.1, p. 223], we find that the Jacobian of X is isogenousto E − × E × E . (To see this, we must take the elliptic curves mentioned inLigozat’s proposition and compute their reductions modulo p .) It follows that over F , the Jacobian of X is isogenous to E a − E b − E E − b E − a for some choice of a and b , and by Remark 3.4, we know that ( a, b ) is one of (4 , , , F -rational points on a curve with that zeta function.We find that if the number of F -rational points on X is 1, then ( a, b ) = (4 , X ( F ) = 3 then ( a, b ) = (5 , X ( F ) = 5 then ( a, b ) = (6 , E , the unique elliptic curve over F with trace 0. The characteristicpolynomial of Frobenius on E [11] is x + 2, so the action of Frobenius on E [11]has two eigenspaces, one with eigenvalue 3 and one with eigenvalue −
3. One findsthat there are 10 pairing-respecting isomorphisms ϕ : E [11] → G × G ′ , and since( E , ϕ ) and ( E , − ϕ ) are represented by the same point on X , we have found 5 F -rational points on X . Therefore the Jacobian of X decomposes as claimed inthe statement of the theorem. (cid:3) Remark . One can show that the curve X has 60 cusps (that is, points that lieover the point at infinity on the the j -line). The field of definition of 10 of the cuspsis F ; the field of definition of the other 50 is F . Using these facts, togetherwith the modular interpretation of the non-cuspidal points on X , we can computethe number of points on X over any (reasonably small) extension of F . This givesanother method of computing the decomposition of the Jacobian of X . Remark . Applying Ligozat’s Proposition 3.6.1, we see that the Jacobian of X (11) K ⊗ F splits into a product of ordinary elliptic curves. Hence the bound26 for ordinary elliptic curves over F from Remark 3.7 is reached as well.5. Application: Modular curves with split Jacobians
Let J ( N ) denote the Jacobian of the modular curve X ( N ) over Q . Cohen [3](mentioned in [15, Remarque 2, p. 90] with the value N = 27 omitted) has computeda list of the odd integers N for which J ( N ) is isogenous to a product of ellipticcurves, and Yamauchi [21, Thm. 1.1] extended this list to include even values of N as well. In this section we use Theorem 3.1 and the mathematical software packageSage [17] to recompute Yamauchi’s list; we note that the list in Yamauchi’s theoremmistakenly includes N = 672 and omits N = 28. Theorem 5.1.
Let N = 2 e n with n odd. Then J ( N ) is isogenous over Q to aproduct of elliptic curves if and only if n appears in the following table and e ≤ E ( n ) with E ( n ) as tabulated : n E ( n ) n E ( n ) n E ( n )1 7 15 4 37 03 7 17 1 45 45 4 19 2 49 07 4 21 4 57 19 7 25 4 75 411 2 27 4 99 213 2 33 2 121 0 ENUS BOUNDS FOR CURVES 13
Proof.
Suppose N is an odd integer such that J ( N ) splits into elliptic curves.Since N is odd the modular curve X ( N ) has good reduction modulo 2, and thereduced curve over F has split Jacobian. By Theorem 3.1, the genus of X ( N ) isat most 26. Using the fact [4] that the genus of X ( N ) is greater than or equal to( N − √ N − /
12, we find that N is at most 422.Using the command J0(N).decomposition() of the mathematics package Sagefor all N less than 423, we find that the odd values of N with J ( N ) split are thetwenty-one odd integers that appear as n in the table.To complete the proof, we note that if J ( N ) is split then so is J ( n ) for everydivisor n of N . Therefore the integers we are searching for can be written 2 e n forsome exponent e and for some n among the odd values that we have just computed.For each of the possible odd parts n , we use Sage to compute the decompositionof J (2 e n ) for increasing values of e until we reach a Jacobian that does not split.(In practice, we do not have to compute the decomposition of J (2 e n ) if we alreadyknow that J (2 e m ) does not split for some divisor m of n . For example, since J (2 )does not split we must have E ( n ) < n .) The largest value of e for which J (2 e n ) splits is recorded in the table as E ( n ). (cid:3) Remark . Ekedahl and Serre [9] give a list of various values of g such that thereexists a curve of genus g over Q with a completely split Jacobian. Many of theirvalues of g come from modular curves X ( N ). The tables of modular forms thatthey had access to did not include values of N greater than 1000, so they missed afew of the values from Theorem 5.1. The modular curves X (1152) and X (1200),of genus 161 and 205, respectively, allow us to add two more values of g to theirlist [9, Th´eor`eme, p. 509].Ekedahl and Serre also seem to have missed the curve X (396) of genus 61, butthey obtain a curve of genus 61 with split Jacobian by considering a quotient of X (720) by an involution. References [1] Peter Bending, Alan Camina, and Robert Guralnick,
Automorphisms of the modular curve ,Progress in Galois theory, Dev. Math., vol. 12, Springer, New York, 2005, pp. 25–37.DOI: 10.1007/0-387-23534-5 2.[2] Wieb Bosma, John Cannon, and Catherine Playoust,
The Magma algebra system. I. Theuser language , J. Symbolic Comput. (1997), no. 3-4, 235–265. Computational algebra andnumber theory (London, 1993). DOI: 10.1006/jsco.1996.0125.[3] Henri Cohen, Sur les N tels que J ( N ) soit Q -isog`ene `a un produit de courbes elliptiques On the genera of X ( N ) (2000).arXiv:math/0006096v2 [math.NT].[5] Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenk¨orper , Abh. Math.Sem. Hansischen Univ. (1941), 197–272. DOI: 10.1007/BF02940746.[6] Stephen A. DiPippo and Everett W. Howe, Real polynomials with all roots on the unit cir-cle and abelian varieties over finite fields , J. Number Theory (1998), no. 2, 426–450.DOI: 10.1006/jnth.1998.2302.[7] , Corrigendum: “Real polynomials with all roots on the unit circle and abelian varietiesover finite fields” , J. Number Theory (2000), no. 1, 182. DOI: 10.1006/jnth.2000.2565.[8] Iwan Duursma and Jean-Yves Enjalbert, Bounds for completely decomposable Jacobians ,Finite fields with applications to coding theory, cryptography and related areas (Oaxaca,2001), Springer, Berlin, 2002, pp. 86–93. Electronic version, with an addendum, available atarXiv:1007.3344v1 [math.NT]. [9] Torsten Ekedahl and Jean-Pierre Serre,
Exemples de courbes alg´ebriques `a jacobiennecompl`etement d´ecomposable , C. R. Acad. Sci. Paris S´er. I Math. (1993), no. 5, 509–513. [=Serre Œuvres 159].[10] Daniel Goldstein, Robert M. Guralnick, Everett W. Howe, and Michael E. Zieve,
Noniso-morphic curves that become isomorphic over extensions of coprime degrees , J. Algebra (2008), no. 6, 2526–2558. DOI: 10.1016/j.jalgebra.2008.06.003.[11] Yasutaka Ihara,
Some remarks on the number of rational points of algebraic curves overfinite fields , J. Fac. Sci. Univ. Tokyo Sect. IA Math. (1981), no. 3, 721–724 (1982). http://hdl.handle.net/2261/6319 .[12] G´erard Ligozat, Courbes modulaires de niveau
11, Modular functions of one variable, V (Proc.Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1977, pp. 149–237. LectureNotes in Math., Vol. 601. DOI: 10.1007/BFb0063948.[13] Christophe Ritzenthaler,
Automorphismes des courbes modulaires X ( n ) en caract´eristique p ,Manuscripta Math. (2002), no. 1, 49–62. DOI: 10.1007/s002290200286.[14] Hans-Georg R¨uck and Henning Stichtenoth, A characterization of Hermitianfunction fields over finite fields , J. Reine Angew. Math. (1994), 185–188.DOI: 10.1515/crll.1994.457.185.[15] Jean-Pierre Serre,
R´epartition asymptotique des valeurs propres de l’op´erateur deHecke T p , J. Amer. Math. Soc. (1997), no. 1, 75–102. [=Œuvres 170]DOI: 10.1090/S0894-0347-97-00220-8.[16] C. J. Smyth, Some inequalities for certain power sums , Acta Math. Acad. Sci. Hungar. (1976), no. 3–4, 271–273. DOI: 10.1007/BF01896789.[17] William Stein et al., Sage Mathematics Software (Version 4.0.1) , the Sage DevelopmentTeam, 2009. .[18] Michael A. Tsfasman,
Some remarks on the asymptotic number of points , Coding theory andalgebraic geometry (Luminy, 1991), Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992,pp. 178–192. DOI: 10.1007/BFb0088001.[19] M. A. Tsfasman and S. G. Vl˘adut¸,
Asymptotic properties of zeta-functions , J. Math. Sci.(New York) (1997), no. 5, 1445–1467. Algebraic geometry, 7. DOI: 10.1007/BF02399198.[20] William C. Waterhouse, Abelian varieties over finite fields , Ann. Sci. ´Ecole Norm. Sup. (4) (1969), 521–560. .[21] Takuya Yamauchi, On Q -simple factors of Jacobian varieties of modular curves , YokohamaMath. J. (2007), no. 2, 149–160. Department of Mathematics, Harvard University, Cambridge, MA 02138–2901
E-mail address : [email protected] URL : Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967, USA.
E-mail address : [email protected] URL : Institut de Math´ematiques de Luminy, UMR 6206 du CNRS, Luminy, Case 907, 13288Marseille, France.
E-mail address : [email protected] URL ::