Abstract
We give a method for explicitly constructing an elementary cubic extension over which an elliptic curve has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of and a generic polynomial for cyclic cubic extensions. In our method, the extension degree often becomes small.
1. Introduction
Let be an elliptic curve defined over a number field . It is well-known that the Mordell-Weil group of -rational points on forms a finitely generated abelian group, and its is of great interest in arithmetic geometry. The present paper is motivated by the following general problem:
To understand the behavior of ranks of elliptic curves in towers of finite extensions over .
This problem dates back at least to [1], which first introduced an Iwasawa theory for elliptic curves. In the context of Iwasawa theory, Kurcanov [2] showed that if an elliptic curve over without complex multiplication has good reduction at a prime number satisfying a mild condition then has infinite rank in a certain -extension, and Harris [3] showed that if is a modular elliptic curve over having good ordinary reduction at and a specific -rational point arising from the modular curve then there is a -adic Lie extension of in which the rank of grows infinitely. On the other hand, Kida [4] constructed a tower of elementary -extensions in which the rank of becomes arbitrarily large by using a result in theory of congruent numbers. Also, Dokchitser [5] proved that for any elliptic curve over there are infinitely many cubic extensions so that the rank of increases in . In the present paper, we consider the specific curve and give an explicit construction of an elementary cubic extension over which the elliptic curve is of rank of at least a given positive integer by finding a close connection between a -isogeny of and a generic polynomial for cyclic cubic extensions. We shall call an extension an elementary cubic extension if is a finite compositum of cubic extensions over . Compared with related results as above, in our method, the extension degree often becomes small. The question of finding a small elementary cubic extension with rank seems to be of independent interest.
2. Main Results
Let We denote by a fixed algebraic closure of . Let denote the field obtained by adjoining a real number for each to . Since the set is finite, the extension is also finite. The main result of the paper is the following theorem.
Theorem 1. Let be any square-free product of distinct primes satisfying . Then, for any positive integer , we can construct a subfield of such that
As we will see in Section 4, the field in Theorem 1 is effectively computable from the decomposition of the primes in , where denotes a primitive cube root of unity. See Remark 4 for this computation. It will also turn out in Section 4 that the field is constructed using certain products of prime elements in above each prime , and in particular depends only on the primes and a positive integer . One question arises here: how small can be when vary through the set of all primes for a fixed integer ? We give some examples in the case (Table 1).
Corollary 2. Let be any square-free product of distinct primes satisfying . Then, for any positive integer , we can construct a subfield of with some integer such that
Note that our construction in Theorem 1 (resp., Corollary 2) differs from Dokchitser’s [5], and the extension degree (resp., ) is often smaller than or equal to (resp., ).
3. A Connection between the Elliptic Curve and a Generic Polynomial for Cyclic Cubic Extensions
In this section, let be a perfect field with characteristic not equal to , and let be any element in . The elliptic curve has a rational -torsion point and admits a -isogeny over . The -isogeny can be explicitly written asThe isogenous elliptic curve is given by the form Taking Galois cohomology of the short exact sequence yields a Kummer map Combining this map with the map where denotes the set of all cyclic cubic extensions over together with , we have a map
It is well-known that the set is parametrized by a generic polynomial for -extensions [6]which has discriminant . We shall denote by the minimal splitting field of over , where stands for the prime field in . By the property of , for any there is some so as to be in .
The following proposition is the key observation in the present paper.
Proposition 3. The connecting homomorphism is given by for any . Furthermore, in the case where is a number field, for any prime ideal of not dividing , the -component of the conductor of isHere denotes the normalized valuation of the -adic completion .
Proof. For any , take a point on satisfying . By the form of the -coordinate of (5), the splitting field coincides with . Here . The latter part of the proposition follows from the equation by using Proposition 8.1 in [7].
4. A Method for Constructing an Elementary Cubic Extension
From now on, let be a square-free product of distinct primes satisfying , and let be a positive integer .
Recall a result of Komatsu [8] that the set of all cyclic cubic fields (i.e., cyclic cubic extensions over ) of conductor has a one-to-one correspondence to the subset of the algebraic torus consisting of elements of the form where is the composition of given by , denotes the inverse of given by , and is a rational number so that the pair of integers is unique in the sense of Lemma 2.1 in [8] (especially , which will be used in the following) and satisfies There is a group isomorphism mapping to , where denotes the multiplicative group. Then the composition is written as .
Now, let , which is a prime element in dividing , and let denote the complex conjugation of . For each , let Since for each , there exist unique integers , so that where is relatively prime to . Then , and thus .
Let .
Remark 4. In order to calculate , we have only to know the values by the definition of . For example, one can use Table 5.1 in [8], in which the values for are listed. From the construction of , the field depends only on and a positive integer .
Lemma 5. Every prime number is unramified in .
Proof. Since does not divide and is unramified in , it turns out that is unramified in by Kummer theory. Hence, is also unramified in .
Proposition 6. for any .
Proof. Since , we haveThus . It follows from Proposition 3 that , and hence . Combining Proposition 3 with Lemma 5 yields .
Proposition 7. .
Proof. One can easily verify that is a -torsion point in , and the rational function on has a zero of order at and a pole of order at infinity and satisfies Then there exists an injective homomorphism (see [9] for details): Combining this with the natural mapwe have the homomorphismBy Proposition 6, the point corresponds to via the above homomorphism. It thus suffices to show that is linearly independent in . Then the points must be linearly independent in . Assume that for some integers . For each , let be a prime ideal of above the prime element . For any prime ideal of , we denote by the normalized (additive) valuation of the -adic completion . By Lemma 5, we have . Thenand hence for each . This proves that is linearly independent in .
Let be the dual isogeny to . Since the composition (resp., ) is the multiplication-by- map on (resp., ), we have an exact sequence of -vector spaces and thus Since , the group is trivial and the group is generated by the point . We see that
Proof of Theorem 1. By Proposition 7, it suffices to show that . Then Assume that there exists some point such that . Then . It follows from the -division polynomial for that there is no nontrivial -torsion point in . Therefore, must be the trivial element in . This contradicts the assumption, and hence .
Proof of Corollary 2. By Theorem 1, we have only to consider the case . (In the case where , take .) Take distinct prime numbers not dividing with . Let . Then, applying Theorem 1 to the elliptic curve , there exists some subfield so as to be with . Let . Then, it is easily seen that is isomorphic over to . Therefore, . Here .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author is supported by APU Academic Research Subsidy.