Research Article | Open Access
On the Rank of Elliptic Curves in Elementary Cubic Extensions
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.
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 , which first introduced an Iwasawa theory for elliptic curves. In the context of Iwasawa theory, Kurcanov  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  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  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  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
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 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 .
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  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  (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 .
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 , 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 .
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  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.
The author is supported by APU Academic Research Subsidy.
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Copyright © 2015 Rintaro Kozuma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.