A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM

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1 A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM LARRY ROLEN Abstract. We study a certain generalization of the classical Congruent Nuber Proble. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle θ. These nubers are called θ-congruent. We give an elliptic curve criterion for deterining whether a given integer n is θ-congruent. We then consider the density of integers n which are θ-congruent, as well as the related proble giving the density of angles θ for which a fixed n is congruent. Assuing the Shafarevich-Tate conjecture, we prove that both proportions are at least 50% in the liit. To obtain our result we use the recently proven p-parity conjecture due to Monsky and the Dokchitsers as well as a theore of Helfgott on average root nubers in algebraic failies. 1. Introduction and Stateent of Results The study of right triangles with integer side lengths dates back to the work of Pythagoras and Euclid, and the ancients copletely classified such triangles. Another proble involving triangles with nice side lengths was first studied systeatically by the Arab atheaticians of the 10 th Century. This proble asks for a classification of all possible areas of right triangles with rational side lengths. A positive integer n is congruent if it is the area of a right triangle with all rational side lengths. In other words, there exist rational nubers a, b, c satisfying a + b = c ab = n. The proble of classifying congruent nubers reduces to the cases where n is square-free. We can scale areas trivially. We can easily generate exaples of congruent nubers; for exaple 6 is congruent and is given by the triangle. Classically, people were able to solve this proble in a few cases using exaples and eleentary techniques. For exaple, Ferat proved that 1 is not a congruent nuber and Euler was the first to find a triangle showing that 7 is. Given an arbitrary integer, however, it traditionally seeed hopeless to find a siple, general algorith to test for congruency. To deonstrate the potential coplexity of finding such triangles in general, consider the following exaple, due to Zagier. He coputed the siplest right triangle representing 157 as congruent. The author is an undergraduate at the University of Wisconsin ajoring in atheatics. He is grateful for the support provided by the NSF Research Training Grant in Nuber Theory (PI: Ono). He would also like to thank Ken Ono for his guidance and support. He also thanks David Brown for providing coents on an earlier draft of this paper. 1

2 LARRY ROLEN In this triangle, the hypotenuse is given by For ore on the history of the congruent nuber proble see [4]. The natural way to view this proble is in ters of rational points on elliptic curves. Naely, to every n we associate a corresponding elliptic curve E n whose Mordell-Weil group encodes the data of rational right triangles of area n. For each square-free positive integer n, we define the congruent nuber curve E n : y = x n x. There is a 1 correspondence between rational right triangles of area n and rational points (x, y) on E n with y 0. Now E n clearly has full -torsion as it contains the points O, (0, 0), ( n, 0), and (n, 0). It turns out that these are all of the torsion points; i.e En tors = Z/Z Z/Z. As a corollary, n is congruent if and only if rankq E n > 0. Using this characterization along with the Hasse-Weil L-function and theory of half-integral weight odular fors, Tunnell was able to derive the a rearkable classification of congruent nubers. Assuing the weak Birch and Swinnerton-Dyer (BSD) conjecture, he gives an equivalent condition for a nuber n to be congruent in ters of the nuber of representations of n by two quadratic fors; this condition ay be checked as quickly as in tie O(n 1 ) (for exaple, see [1][15].) We would like to consider a natural generalization of the congruent nuber proble. Naely, we will relax the condition that the triangle have a right angle and consider ore general angles. In analogy with the previous definition, let π θ π be an angle. 3 We say that a square-free integer n is θ-congruent if there exists a triangle whose largest angle is θ, whose side lengths are rational, and whose area is n. In other words, there exist rational a, b, c for which a + b ab cos θ = c ab sin θ = n b c θ a We say that an angle π/3 θ π is adissible if both sin θ and cos θ lie in Q. Following the classical paraeterization of rational points on the unit circle, define to be the slope of the line joining ( 1, 0) to (cos θ, sin θ). Call a rational nuber adissible if it corresponds to an adissible angle. Note that = 1 corresponds to θ = π/. Then the rational points on the circle are in one-to-one correspondence with rational choices of. It is easy to derive the forulae: cos θ = sin θ = 1 +. Henceforth the dependence of on θ will be iplicit. Thus, for every such rational > 3 we have a separate congruent nuber proble, with = 1 corresponding to 3 the classical congruent nuber proble. We are alost in position to give an elliptic curve criterion for the generalized congruent nuber proble. First we introduce soe language which pertains to the special case of certain angles. We say that an adissible Q is aberrant provided that + 1 Q. Otherwise, an adissible is called

3 A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM 3 generic. We can paraeterize all aberrant as follows. Begin with the paraeterization of Pythagorean triples of Euclid which states that any priitive triple is of the for (u v, uv, u + v ) for relatively prie (u, v). It easily follows that all aberrant ( ) ±1 can be written in the for for relatively prie integers u, v. u v uv To each aberrant, we can associate a (unique) square-free natural nuber n such that n Q and we call the pair (n, ) aberrant as well. Any other pair is tered generic. To any adissible pair (n, ) we associate the elliptic curve E n,θ given by the following Weierstrass equation: ( (1) E n,θ : y = x x n ) (x + n). Using this equation it is ore natural to paraeterize angles by the rational than an angle θ. For exaple, there is a duality between and 1 in the aberrant case which is uch less transparent in the θ-characterization. There are two special properties characterizing aberrant nubers which propt their nae. Note that there are no congruent equilateral triangles. The following two theores give the first exaple of the special aberrant behavior as well as our first exaple of a situation which cannot occur in the classical congruent nuber case (as is irrational): Theore 1.1. If (n, ) is aberrant, then n is a θ -congruent nuber and n can be represented by an isosceles θ -triangle. In fact, we can give a characterization of all isosceles triangles appearing in the generalized congruent nuber proble. Theore 1.. All isosceles triangles with rational side lengths and areas correspond to the aberrant case. One property of generalized congruent nuber case which reains the sae as the θ = π case is the following correspondence which translates our proble into one of coputing Mordell-Weil groups. Theore 1.3. For any positive square-free integer n and any adissible angle θ we have that n is θ-congruent if and only if E n,θ has a rational point (x, y) with y 0. Moreover, if (n, ) is generic there is a -1 correspondence of triangles representing the pair (n, ) and such points on E n,θ. In fact, (x, y) and (x, y ) correspond to the sae triangle if and only if x = x and y = ±y. In order to use the elliptic curve efficiently in our study we ust first copute the torsion subgroup of E n,θ. This will allow us to give the desired equivalent condition in ters of Mordell-Weil ranks. We prove the following classification: Theore 1.4. If (n, ) is aberrant, then En,θ tors (Q) = Z/Z Z/4Z. If (n, ) is generic, then En,θ tors (Q) = Z/Z Z/Z.

4 4 LARRY ROLEN Corollary 1.5. We have that (n, ) is a congruent pair if and only if (n, ) is aberrant or rank Q E n,θ (Q) > 0. In this paper, we would like to analyze a different aspect of the proble fro that considered by Tunnell. Although it reains a difficult proble to deterine general conditions to test for congruent nubers (even parts of the classical congruent nuber proble resolution reain conjectural), we are still able to ake general stateents about the distribution of such points. There are two natural failies to consider along with the following questions. Question 1. If we fix and let the area n vary, how often is (n, ) congruent? Question. If we fix the area n and let vary, how often is (n, ) congruent? For each area n, there is at least one angle aking n a θ -congruent nuber. To see this, note that ( n+, n 4) is a rational point on E 4 n, n with non-zero y-coordinate 4 unless n =. For n =, (9/, 15) is a rational point on E,4 so is 4-congruent. A natural question is if for each n there are infinitely any angles which ake n congruent. Even better, we would like to say as uch as we can pertaining to approxiately how often nubers are congruent or not. To this end, we let h (x) be the proportion of positive square-free integers n not exceeding x for which (n, ) is a congruent pair and v n (x) the proportion of with height at ost x for which (n, ) is a congruent pair. More precisely, () h (x) := #{1 n x, : n is θ -congruent and n is square-free} #{1 n x : n is square-free} (3) v n (x) := #{ Q : h() n and n is θ -congruent} #{ Q : h() n} where for any rational nuber written in lowest ters h( a ) := ax{ a, b }. Then we b prove the following: Theore 1.6. Suppose the Shafarevich-Tate conjecture is true for all elliptic curves of rank 0. Then for each ɛ > 0, if x ɛ 0 then 1 ɛ h (x) < 1 ɛ That is, when we fix the angle and let the area n vary, at least 1/ of the choices for n are congruent and a positive density are non-congruent. Fixing n and letting vary, we also obtain the following: Theore 1.7. Suppose the Shafarevich-Tate conjecture is true for all elliptic curves of rank 0. Then for each ɛ > 0, if x ɛ 0 then 1 ɛ v n(x) 1 ɛ. The paper is organized as follows. In we will prove Theores , establishing the equivalent condition for congruence in ters of ranks of elliptic curves. In 3, we prove the density results Theore 1.6 and Theore 1.7. These rely on a theore of Yu for quadratic twists giving the upper bound in Theore 1.7 and a rearkable new

5 A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM 5 result of the Dokchitser brothers which proves the p-parity conjecture in the case when p is odd (the case where p = being previously solved by Monsky) in conjunction with a paper of Helfgott providing sufficient conditions for algebraic failies of elliptic curves to have average root nuber zero [16],[5],[8],[11].. Proof of the Elliptic Curve Criterion and Coputation of Torsion Subgroups for E n,θ. In this section, we prove the theores necessary to reduce the generalized congruent nuber proble to one of coputing the rank of E n,θ. We begin by establishing the connection between E n,θ (Q) and congruent nubers..1. Connecting E n,θ to θ -Congruent Nubers. We begin with a proof of Theore 1.1. Suppose (n, ) is aberrant. Then as n is a square, so is n. It is siple to n( check that the triangle with side lengths a = b = +1), c = n has angle θ and area n. To finish the characterization of isosceles congruent nuber triangles in Theore 1., suppose that a = b. By the law of cosines ( ) c = a a = 4a. + 1 We can write + 1 = ( a c ). Now by the area forula we can rewrite n = a, 1+ establishing the aberrance of (n, ). We proceed to establish the elliptic curve condition in the generic case as per Theore 1.3. Thus, suppose n is θ -congruent. Then define (x, y) := ( c, b c a c). In the generic 4 8 case, the triangle cannot be isosceles by Theore 1. and hence a b giving y 0. It is trivial to verify that this point indeed lies on E n,θ. Conversely, given a rational point (x, y) with non-zero y coordinate, define a triangle with side lengths a = n( x + x) y, b = (x + n)(x n) y, c = x + n y. Note that if (n, ) is an aberrant pair, then by Theore 1.4 the torsion subgroup is Z/Z Z/4Z. This is aberrant by Theore 1.1 and evidently contains a torsion point with y 0, proving Theore Coputing the Torsion Subgroups. In this section we copute the torsion subgroup of E n,θ : y = x(x n )(x + n) = x3 + n( 1) x n x. First choose 0 α Z such that α n, α n 3 are both integral. Then by the change of variables (x, y) ((α) x, (α) 3 y) this curve is isoorphic (over Q) to y =

6 6 LARRY ROLEN x 3 + α n( 1)x α 4 4 n x. If we define M := n 3 α and N := nα, we can write the curve in the for y = x 3 + (M + N)x + MNx with M and N integral. Now there is a coplete characterization of torsion subgroups of curves of this for in [1]. Note that they all have full -torsion, so by Mazur s Theore the torsion subgroup is of the for Z/Z Z/nZ for n {1,, 3, 4}. Theore.1. (Ono [1]). The following criteria uniquely deterine the torsion subgroups of E(M, N) : y = x 3 + (M + N)x + MNx. E(M, N) tors contains Z/Z Z/4Z if M and N are both squares, or M and N M are both squares or N and M N are both squares. E(M, N) tors = Z/Z Z/8Z if there exists a non-zero integer d such that M = d u 4 and N = d v 4, or M = d v 4 and N = d (u 4 v 4 ), or M = d (u 4 v 4 ) and N = d v 4 where (u, v, w) fors a Pythagorean triple (i.e. u + v = w ). E(M, N) tors = Z/Z Z/6Z if there exist integers a, b such that a b {, 1, 1 and M = a 4 + a 3 b and N = ab 3 + b 4. Otherwise, E(M, N) tors = Z/Z Z/Z. We continue by checking the conditions outlined there. To check if the torsion subgroup contains Z/Z Z/4Z, note first that neither N nor M can be a square as both n, are positive. Thus, the torsion subgroup will have a 4-torsion point if and only if N = nα and M N = α n( + 1) are both squares. This is clearly equivalent to the aberrant condition. Thus, in the generic case, there can be no 4-torsion. Let us rule out the possibility of 8-torsion. Suppose the torsion subgroup is Z/Z Z/8Z. Then again by sign considerations neither N nor M are squares and so this happens if and only if we can write M = d (u 4 v 4 ) and N = d v 4 for a non-zero integer d and a Pythagorean triple (u, v, w). In this case, we have M = N = v4 u 4 = 1 ( u v 4 v )4 and so 1 + is a rational fourth power. I clai that this is not possible. For suppose 1 + ( c d ) = ( e f )4 with c, d, e, f integral. Then (f c) + (f d) = (e d) and so f 4 c = d (e 4 f 4 ). But this forces e 4 f 4 to be a perfect square g and hence gives a Pythagorean triple (g, f, e ), which is ipossible as it is well-known that a Pythagorean triple can contain at ost one square (as was shown by Ferat). To finish the characterization of the torsion subgroups, we have only to exlude the possibility Z/Z Z/6Z. Suppose we have a, b as in the Theore. In this case, if we have M = a a +ab = N b b +ab and so we can write a +ab = c for c, d Z. By rearranging and considering this as a quadratic equation in the variable a, by the quadratic b +ab d forula the discriinant ust be integral and this gives that c 4 + d 4 c d ust be a perfect square. This is a special case of a well-studied class of equations. In particular, Diophantine equations of the for x 4 + nx y + y 4 = z have been studied intensely for over 300 years. Ferat s proof that there are no nontrivial integral solutions when n = 0 is a classic exaple of infinite descent, and when n = 1 this equation is central to the well-known proof that four perfect squares cannot lie in arithetic progression. The only integral solutions occur when cd = 0 or c = d. An eleentary proof of this, 0, 1}

7 A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM 7 fact ay be found in [13]. Then if cd = 0, d 0 and so c = 0 giving = 0, which is a contradiction. In the latter case, we also have that a = b and so = 1 giving = 1. But the case = 1 was already proven to have no 3-torsion in Lea 1, being the classical congruent nuber case. 3. Proof of Density Results In this section we prove Theores 1.7 and 1.8 describing the average behavior of congruent nubers in one-paraeter failies. This will be done by coputing averages of root nubers and using the parity conjecture A Sufficient Condition for Lower Bounds. In order to describe the proof, we first need to discuss a faous conjecture on the parity of ranks of elliptic curves. Let us recall that every elliptic curve has an associated Hasse-Weil L-function which provides the connection to odular fors in the Modularity Theore of Wiles, Taylor, et. al and whose value at s = 1 conjecturally (BSD) gives the rank of E. For each prie p, let N(p) be the nuber of points (including the point at infinity) of the reduced curve E p. Then set a(p) := p + 1 N(p). Finally, define the L-series by the Euler product: L(E, s) := n=1 a(n) n s = p a(p)p s 1 a(p)p s + p, 1 s where denotes the inial discriinant of E. This L-function can be analytically continued to an entire function, as proven by Wiles et al. It also has a special syetry which is expressed in the functional equation: N s E (π)s Γ( s)l(e, s) = W (E)N s E (π) s Γ(s)L(E, s). Here N E is the conductor of E and W (E) = ±1 is the root nuber of E. It is expected that the root nuber controls the parity of the rank of E. Specifically, the following is also an iediate consequence of BSD: Conjecture. (Parity) For any elliptic curve E/Q, W (E) = ( 1) rk Q(E) In certain cases, the parity conjecture ay be shown to be true unconditionally. In particular, a powerful new result of Doskchitser proving the p-parity conjecture has the following corollary: Theore 3.1. (Corollary 4.0 of Dokchitser [5]) If E/Q is an elliptic curve, either the parity conjecture is true for E or X(E/Q) contains a copy of Q/Z. Here X(E/Q) is the group of hoogenous spaces for E/Q which have a solution in every copletion of Q odulo equivalence. Together with the Seler group it easures how badly the Hasse principle fails for E (i.e. the inability to lift solutions over every copletion to a solution over the global field). As a corollary to Theore 3.1, we now state our sufficient condition for the density lower bounds in Theores 1.6 and 1.7. p

8 8 LARRY ROLEN Theore 3.. If a faily of elliptic curves over Q has average root nuber 0 and the Shafarevich-Tate Conjecture is true for elliptic curves of rank 0, then the proportion of rank 0 curves in this faily is at ost 1. Proof. If X(E)/Q is finite and rank(e(q)) is zero, then E will have positive root nuber by the previous theore. Given that the average root nuber is zero, the total proportion of positive root nuber curves is 1 and so the proportion of rank 0 curves cannot exceed this aount. 3.. Proof of Theore 1.6. We are now in position to prove the bounds on the proportion of quadratic twists with positive rank stated in Theore 1.6. For the lower bound, we recall the well-known fact that the average root nuber in failies of quadratic twists is 0. For the upper bound, we apply a theore of Yu. This is proven by coputing the average -Seler group size. The result follows fro the Dokchitsers result and the proof of the p-parity conjecture for the case p = [11]. Theore 3.3. (Yu [16]) Let E/Q be an elliptic curve with torsion subgroup containing Z/Z Z/Z. Then a positive proportion of quadratic twists have rank 0. By Theore 1.4, our curves always satisfy this condition as (, 1) is never aberrant by the paraeterization of all aberrant pairs stated in the introduction Proof of Theore 1.7. To prove the lower bound on the proportion of positiverank curves in the faily of curves obtained by fixing the area n, we need a ore general theore describing average root nubers in elliptic fibrations. Although in our case the following conjectures will hold unconditionally, let us state the following hypotheses which relate to classical arithetic conjectures. Hypothesis. (A) Let P (x, y) be a hoogenous polynoial. Then only for a zero proportion of all pairs of coprie integers (x, y) do we have a prie p > ax{x, y} such that p P (x, y). It is believed that this is true for all such square-free P. In particular, this is iplied by the abc-conjecture [7]. Fortunately in our case, it has been proven unconditionally true whenever P has no irreducible factor of degree exceeding 6 [10]. Another hypothesis we will need is: Hypothesis. (B) Let λ(n) := p n ( 1) νp(n) be the Liouville function. Then λ(p (x, y)) has strong zero average over Z. Chowla has conjectured that this holds for all non-constant, square-free hoogenous polynoials P []. The prie nuber theore is essentially equivalent to the case where deg(p ) = 1, and it has been proven unconditionally whenever deg(p ) = 1,, 3 [9].

9 A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM 9 Now if we have an elliptic curve E over Q(t), we can for two polynoials fored by a product over the places of bad reduction based on type: M E := P ν, B E := P ν. E has ult. red. at ν E has q. bad red. at ν Here P ν := y if ν is the infinite place and otherwise P ν := y deg(q) Q( x ) for the irreducible y polynoial Q inducing ν. We also say a curve has quite bad (q. bad) reduction at a place if every quadratic twist also has bad reduction at the sae place. We are ready to state the sufficient conditions for coputing our average of root nubers: Theore 3.4. (Helfgott [8]) Let E be an elliptic curve over Q(t). Suppose M E 1 (i.e. E has a point of ultiplicative reduction). Suppose further that Hypothesis A holds for B E and Hypothesis B holds for M E. Then the strong average over Q of W (E t ) of the fibres exists and is 0. Let us observe how this applies to our case. Our faily of elliptic curves with varying is an elliptic curve over Q(t) with t =. First we calculate the usual constants associated to this curve: c 4 = 16n ( + 1)( + + 1)), c 6 = 3n3 ( 1)( + 1)( + )( + 1) 3 = 16n6 ( + 1) Thus, the curve has bad reduction at three places: {,, + 1}. By checking the valuation criteria for reduction type, we find that M E = x + y and B E = xy(x + y ). Fro the above rearks, the required hypothesis on M E, B E hold unconditionally and so Helfgott s theore applies to show the strong average of root nubers is zero. The exact stateent of what strong average eans is coplicated, but it is a certain notion of average which does not depend on which intervals in Q we saple or how we add ters (a precise definition ay be found in Helfgott). In particular it is clearly a strict enough notion to give us the conditions we need to apply our Lea and conclude the truth of Theore Discussion of Conjectural Densities. Although we are only able to bound the densities as in Theores 1.6 and 1.7, in fact uch ore is believed to be true. For exaple, we have the following well-known hypothesis Conjecture. (Goldfeld [6]) For any faily of quadratic twists, the proportion of curves with rank 0 is 50% and the proportion of curves with rank 1 is 50%. All higher ranks occur only with density 0. In particular, for any faily of quadratic twists and any other reasonable faily of curves such as the second faily with the area n fixed and the angle varying, it is believed that a siilar heuristic should hold. In particular, we ake the following:

10 10 LARRY ROLEN Conjecture 1. For each positive, square-free integer n, (n, ) is not a congruent pair for a positive proportion of angles. For the faily of curves we are considering, the generic rank is 0. The following is a plausible conjecture: Conjecture. (Density). Let E be an elliptic curve over Q(t) and generic rank n. Then only a zero proportion of fibers have rank at least n +. We also say that a curve has elevated rank if only finitely any fibers have generic rank. Thus, our conjecture iplies that the faily of curves does not have elevated rank. For suitably anipulated failies, the rank has been shown to be elevated (see [3] for a discussion of known ethods for constructing exaples). All such proofs rely heavily on the fact that these failies are isotrivial, eaning that the j-invariant j(e) Q does not depend on t. It is believed no non-isotrivial failies can have elevated rank. In particular an arithetic conjecture known as the square-free-conjecture iplies that the average root nuber of a non-isotrivial faily is not ±1 [3]. Thus, our conjecture would follow fro the square-free and density conjectures (assuing the parity conjecture for rank 0 curves). If the full parity conjecture is further assued, there would be no nonisotrivial failies of elevated rank over Q. We also note that, assuing the parity conjecture is true for rank 1 curves, the conjecture would follow if we could bound the average rank strictly by 3. If this could be shown it would constitute the first exaple of a non-quadratic twist elliptic fibration which is known to have average rank not exceeding 3. We can say, assuing a few conjectures such as BSD and the Generalized Rieann Hypothesis, that a Theore of Silveran tells us that the average rank is no larger than 5 [14]. Thus, although the conjecture is alost certainly true, a powerful new theore would be required to say this with certainty. We conclude with a sall table of data illustrating the plausibility of the conjecture as coputed in Sage. Table 1. Ranks for = 1,,..., 100 rank=0 1 3 n= n= n= n= n= References [1] E. Bach and N. Ryan. Efficient Verification of Tunnell s Criterion. Japan J. Indust. Appl. Math., Volue 4, Nuber 3 (007), [] S. Chowla. The Rieann Hypothesis and Hilbert s Tenth Proble. Matheatics and its Applications, Vol. 4 Gordon and Breach Science Publishers, New York-London-Paris, 1965.

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