A result on the analytic -invariant of elliptic curves
We generalise a result by Greenberg and Vatsal on the relation between the analytic -invariants of two elliptic curves whose -torsion subgroups are isomorphic as Galois modules, for suitable .
Let be an elliptic curve over the field of rational numbers and let be an odd prime at which has good ordinary reduction. Mazur and Swinnerton-Dyer [MSD74] attached to the pair a -adic -function , which lies in , where is the ring of power series in the variable with -adic integral coefficients. The function interpolates the central values of the twists of the complex -function of by the characters of the absolute Galois group of factoring through the -cyclotomic extension.
The analytic -invariant of , denoted , is defined as the -adic valuation of the highest power of dividing . Wuthrich [Wut14] proved that is in fact an integral power series, when is a prime of good ordinary reduction: that is, and .
The main result of this article is the following.
Let and be elliptic curves and an odd prime at which both and have good ordinary reduction. Assume that and are isomorphic as Galois modules and that and are irreducible. Then . Furthermore, if and are isomorphic, then .
Theorem 1.1 is a generalisation of a part of[GV00, Theorem (1.4)], which covers the case when . By extending Greenberg and Vatsal’s result to a more general setting, we aim to provide a perhaps more detailed proof of their statement in [GV00] as well. A key ingredient is the theory of canonical periods of cuspidal eigenforms and resulting congruences between algebraic values of -functions developed by Vatsal in [Vat99] and [Vat13].
We remark that Theorem 1.1 already has an algebraic counterpart in the literature. Namely, Barman and Saikia [BS10] proved the analogue of Theorem 1.1 for the algebraic -invariants of and . Here by the algebraic -invariant of we mean the -adic valuation of the highest power of dividing the characteristic series of the Pontryagin dual of the -primary part of the Selmer group of over the -cyclotomic extension of (see [Gre99]). Barman and Saikia’s result is proved under the assumption of triviality of the groups and replacing the stronger irreducibility of and .
We note that the method we use does extend to the case when and are reducible, as long as the unique unramified character appearing in their semisimplification is odd. However, Greenberg and Vatsal showed in [GV00, Theorem (1.3)] that in this case the -invariant is zero.
Acknowledgements. The author is extremely grateful to Andrew Wiles for his time and the precious technical help provided. She would also like to thank Jennifer Balakrishnan for her support and for proof-reading and Thanasis Bouganis for some conversations had on the topic.
2. Notation and auxiliary results
Let be an odd prime and fix once and for all embeddings of into and into the algebraic closure of .
We begin by restricting some definitions and results from [Vat99] and [Vat13] to the case of weight 2 cusp forms (rather than dealing with arbitrary weight). Let be a normalised cuspidal eigenform of weight on , for an integer . Let be the -adic field obtained by completing the number field generated over by the coefficients of at a prime above and let be the ring of integers of . Furthermore, denote by the ring of weight cusp forms on with coefficients in and by the -algebra generated by the Hecke operators acting on . There is a homomorphism , which maps a Hecke operator to the corresponding eigenvalue of . We denote its kernel by and write for the maximal ideal of determined by and . Note, in particular, that if is another normalised eigenform in which satisfies a congruence , then and determine the same maximal ideal .
Endow the ring with the trivial action of the group . Then the first cohomology group of the -module is . Let be the set of parabolic elements of , i.e. the set of matrices in which fix exactly one element of . The parabolic cohomology group of Eichler-Shimura is the subgroup of consisting of those homomorphisms which vanish on .
Besides being a group, also carries the structure of a -module (see [Shi94] or [DI95, p. 116] for an explicit description of the action of double cosets). We denote by the localisation of at . Furthermore, complex conjugation acts on and we write for the -eigenspaces.
There is an such that is cofree of rank , meaning that its dual is a free -module of rank . Such an is called an admissible sign. In symbols,
By duality, we also have an isomorphism as -modules (see [DI95, Proposition 12.4.13]). Therefore, Assumption 1 implies that and we may fix an isomorphism . Denote by the element in the parabolic cohomology group corresponding to under this isomorphism, i.e. .
If for local -algebras , then there is a unique such that factors through and we have (see [Vat13] for more details).
Consider the -valued differential form . The map is a homomorphism , which vanishes on the parabolic subgroup and which is independent of the choice of a basepoint in the upper half plane. Denote by the cohomology class in corresponding to the -cocycle just constructed. Then may be decomposed into a sum of eigenvectors and for the action of complex conjugation. Since is an eigenform of level , the cocycle is an eigenvector also for the Hecke algebra of level . The same is true for . Since is a -dimensional space, there thus exists such that . It is clear that depends on the choice of isomorphism ; however, periods corresponding to different isomorphisms differ merely by multiplication by a -adic unit.
Let be as above. Then is a canonical period attached to and , unique up to multiplication by a -adic unit.
The canonical period of Definition 2.1 corresponds to a choice of transcendental Shimura period. In particular, let be a weight normalised eigenform of level and let be the complex -function of , which is convergent for and admits an analytic continuation to the whole complex plane. If is a Dirichlet character, we also write and (note that where is the trivial character). Then Shimura [Shi76] proved the existence of certain transcendental periods, denoted , with the property that the ratio is an algebraic number. However, the quantity is only defined up to multiplication by an element in . The definition of then corresponds to a choice of Shimura period (up to a -adic unit).
The canonical periods will appear in the normalisation in the interpolation property of the -adic -function of . When is a weight newform corresponding to an elliptic curve , the canonical periods of are related to the Néron periods of . In particular, let be an elliptic curve over with unique global minimal Weierstrass equation
with , and . The invariant differential of this Weierstrass equation for is . Let be the first -homology group of . Then decomposes as a direct sum of two copies of , each being an eigenspace for the action of complex conjugation. In particular, let be a generator of . We define the Néron periods of as .
Let be an elliptic curve over and let be the newform corresponding to by modularity. Let be a prime of good ordinary or multiplicative reduction for and assume that is irreducible. Then and are equal up to multiplication by a -adic unit.
This is [GV00, Proposition (3.1)]. Note that, under the assumption that is irreducible, the Néron periods of the elliptic curves in the isogeny class of are equal up to multiplication by a -adic unit. Thus, the assumption of optimality may be removed from loc. cit.
We briefly sketch the steps in the proof. Denote by the conductor of . Let (resp. ) be the -optimal (resp. -optimal) curve in the isogeny class for and let (resp. ) be a strong (resp. optimal) parametrisation. Then for some (cf. [Ste89, Theorem 1.6]). Similarly, , where is also known to be a -adic unit by a result of Mazur [Maz78]. It is then shown in [GV00, Proposition (3.3)] that and differ by a -adic unit in , from which we deduce what is a -adic unit.
Finally, one considers the map , which sends a -cycle to (note that we have [Shi94, §8.1] and denotes the cap product). Since is normalised, it is in particular non-zero modulo and hence there exists an element such that is a -adic unit. Therefore the image of the integration map defined above is a free -module of rank , generated by . We similarly define a map by , whose image is spanned by . It remains to notice that the images of the two integration maps coincide. ∎
Let be a normalised weight eigenform on . Say that is good ordinary at if and the polynomial has a unique root, say , of vanishing -adic valuation. Let be a Dirichlet character of conductor , where and , with . We define the -adic multiplier of the pair as
and denote by and the Gauss sum and the conductor of .
Let be a topological generator of the Galois group of the -cyclotomic extension of .
Definition 2.4 (The -adic -function of a weight eigenform).
Let be a normalised cuspidal eigenform of weight on , for some integer , and assume that is good ordinary at the odd prime . Let be a Dirichlet character of sign and . Assume that is an admissible sign (cf. Assumption 1). The -adic -function of twisted by is the unique element satisfying the interpolation property
for every root of unity of -power order and where is the Dirichlet character corresponding to the Galois character which maps to .
Similarly, for a finite set of primes not containing we define the -incomplete -adic -function of twisted by as the unique element satisfying the interpolation property
where and are as above and is the -function of twisted by with the Euler factors at the primes in removed.
If is an elliptic curve over and is the corresponding newform, we write for and for .
Let , be as in Definition 2.4; let be a uniformiser for and denote by the -adic valuation on , normalised so that . We define the analytic -invariant of as the -adic valuation of the highest power of dividing if and as otherwise. Given a finite set of primes we also define the analytic -invariant of , denoted , in a similar way.
Let be as in Definition 2.4. Then .
This is a slight generalisation of [GV00, pp. 24-25], where the statement is proved for trivial character . As in [GV00], for a prime , write for the Euler factor at of the -function . Then we have , where and where is such that is the Frobenius automorphism for in . Moreover, for all : if , then it is clear; else is a -adic unit and the argument is the same as for trivial character. ∎
Let be an elliptic curve of level , let the corresponding newform and let be an odd prime. Suppose is good ordinary at and let be an admissible sign for . Let be an eigenform obtained from by deleting all Euler factors corresponding to the primes in a finite set and assume that is also admissible for . Then the canonical periods and are equal up to multiplication by a -adic unit.
We next state a result from [Vat13] (Theorem 3.10) which we will need to establish a congruence between -adic -functions.
Let and be normalised cuspidal eigenforms of weight and of the same level, say , such that . Suppose that, for some integer , and satisfy the congruence , where is a prime of above and that is an admissible sign for (and hence for ). Then, for every Dirichlet character of sign , we have
With the assumptions of Proposition 2.7, suppose further that and are good ordinary at and denote by the ring of integers of . Then
for all -power roots of unity . Therefore
Let respectively be the eigenvalue of resp. for the Hecke operator . Since , in particular we know that . By Hensel’s lemma, this implies that the unique -adic unit root of is congruent modulo to the unique -adic unit root of , which, together with the expression for from Definition 2.4 and with Proposition 2.7, implies the first congruence.
As for the second part of the proof, see [Vat99, Theorem (1.10)]. ∎
For a normalised cuspidal eigenform with rational coefficients, let be the unique semisimple representation satisfying for all . We say that is residually irreducible if is irreducible.
Let be a newform of weight on with rational coefficients, an odd prime such that and suppose that the representation attached to is residually irreducible. Let be a (possibly empty) finite set of primes not containing and let be the eigenform obtained from by removing the Euler factors of corresponding to . Then Assumption 1 holds for for both and .
We have is irreducible. Furthermore, the level of is not divisible by so the claim follows directly from [Wil95, Theorem 2.1 (i)]. ∎
3. Proof of the main result
The theorem is a corollary of the following generalisation of [GV00, Theorem (3.10)].
Let and be elliptic curves of level and and let be a positive integer such that as Galois modules and is irreducible for . Assume that has good ordinary reduction at . Denote by the set of primes dividing . Then
for every Dirichlet character and some . Here .
Denote by the least common multiple of and and by and the newforms corresponding to and by modularity. Furthermore, let denote the Galois representation attached to and similarly for . For all primes at which resp. is unramified, we have
(see e.g. [Rib95, §5]). In particular, this holds at all coprime with . Since the trace of a matrix is an isomorphism-invariant of representations, the hypothesis that , together with the above, implies that for all coprime with .
Denote by the decomposition group at and by the representation of on the -adic Tate module of . Since and are ordinary at by assumption, by [Wil88, Theorem 2] we have
where and are unramified and and . Here is the unique unit root of and is the unique unit root of . Since , we conclude that .
Let respectively be the other root of the corresponding characteristic polynomials of Frobenius. Then . Since and , we also have the congruence . Therefore .
This shows that
Let now and be the normalised cuspidal eigenforms of weight 2 obtained from and by deleting all Euler factors at primes in . These may be viewed as eigenforms of level . By construction, and satisfy the congruence .
We wish to apply Corollary 2.8. In order to do so, we first need to check that is an admissible sign for and . But this follows from the assumption that and are irreducible together with Lemma 2.9. Clearly and are good ordinary at and, with the notation of Corollary 2.8, we have . Therefore, we have a congruence
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.
If , the inequality holds trivially. If , applying Theorem 3.1 with and the trivial character, we obtain the congruence
and since the left hand side is a unit, so is the right hand side. ∎
The assumption of the irreducibility of and is rather strong. In fact, Greenberg conjectured that in this case the algebraic -invariants of and should vanish (see [Gre99, Conjecture 1.11]). Thus the most interesting case of Theorem 1.1 is still probably the one covered in [GV00]. However, Theorem 3.1 also implies a stronger version of Theorem 1.1, as follows.
Let and be elliptic curves, an odd prime at which both and have good ordinary reduction and a Dirichlet character. Assume that and are isomorphic as Galois modules and that and are irreducible. Then . Furthermore, if and are isomorphic, then .
We also note that the statement of Theorem 1.1 remains true if the assumption of good ordinary reduction at the prime is replaced by that of multiplicative reduction. In this case, one appeals to [Wil95, Theorem 2.1 (ii)] to prove that both the newforms attached to and and their modified forms , obtained by deleting all Euler factors at primes dividing , satisfy the cofreeness assumption (Assumption 1). With an appropriate modification of Definition 2.4, one then obtains a congruence between the -adic -functions of and . Finally, an analogue of Lemma 2.6 also holds in this case. We leave the details to the reader.
- [BS10] R. Barman and A. Saikia. A note on Iwasawa -invariants of elliptic curves. Bull. Braz. Math. Soc. (N.S.), 41(3):399–407, 2010.
- [DI95] F. Diamond and J. Im. Modular forms and modular curves. In Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), volume 17 of CMS Conf. Proc., pages 39–133. Amer. Math. Soc., Providence, RI, 1995.
- [Gre99] R. Greenberg. Iwasawa theory for elliptic curves. In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in Math., pages 51–144. Springer, Berlin, 1999.
- [GV00] R. Greenberg and V. Vatsal. On the Iwasawa invariants of elliptic curves. Invent. Math., 142(1):17–63, 2000.
- [Maz78] B. Mazur. Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math., 44(2):129–162, 1978.
- [MSD74] B. Mazur and P. Swinnerton-Dyer. Arithmetic of Weil curves. Invent. Math., 25:1–61, 1974.
- [Rib95] K. A. Ribet. Galois representations and modular forms. Bull. Amer. Math. Soc. (N.S.), 32(4):375–402, 1995.
- [Shi76] G. Shimura. The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math., 29(6):783–804, 1976.
- [Shi94] G. Shimura. Introduction to the arithmetic theory of automorphic functions, volume 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original, Kanô Memorial Lectures, 1.
- [Ste89] G. Stevens. Stickelberger elements and modular parametrizations of elliptic curves. Invent. Math., 98(1):75–106, 1989.
- [Vat99] V. Vatsal. Canonical periods and congruence formulae. Duke Math. J., 98(2):397–419, 1999.
- [Vat13] V. Vatsal. Integral periods for modular forms. Ann. Math. Qué., 37(1):109–128, 2013.
- [Wil88] A. Wiles. On ordinary -adic representations associated to modular forms. Invent. Math., 94(3):529–573, 1988.
- [Wil95] A. Wiles. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2), 141(3):443–551, 1995.
- [Wut14] C. Wuthrich. On the integrality of modular symbols and Kato’s Euler system for elliptic curves. Doc. Math., 19:381–402, 2014.
Mathematical Institute, University of Oxford, Woodstock Rd, Oxford OX2 6GG, UK
E-mail address: firstname.lastname@example.org