The conjecture of Birch and Swinnerton-Dyer connects an analytic invariant of an elliptic curve - the value of the function L, with an algebraic invariant of the curve - the order of the Tate–ˇSafareviˇc group. The assumption is now expressed as a statement about a generator of the image of a map of 1-dimensional modules. To discuss Gross's refinement of the Birch–Swinnerton–Dyer conjecture, we will need the Weil restriction.
The purpose of the Gross conjecture is to identify the leading term of L(ψ, s) rather than that of L(E/F, s) (whose leading term is subject to the Birch–Swinnerton–Dyer conjecture).
Choice of bases
The conjecture
We are interested in the case where rankOKB(K) = 0, after which the assertion of the conjecture reduces to. As explained by Gross, we have the first three norms are the complex norm z 7→ zz, the others are the norm map on ideals), so that the original Birch-Swinnerton-Dyer conjecture follows from Gross's conjecture. In particular, when Q is a Dedekind ring or a field, the determinant can be calculated in this way.
We will use the following implicitly to calculate the determinants of finitely generated torsion moduli.
The setting
Four vector spaces
Galois cohomology
TheKv vector spaceH0(Kv, Bcris⊗QpVp(B)), although invariant under the action of Gal(Kv/Kv), has an additional Frobenius automorphism, which we mean here by φ. Since we are dealing here with the regular rings K,K⊗Qp and OK⊗Zp, the determinants of the complexes under consideration are alternating tensor products of the determinants of the cohomology modules, as indicated in Chapter 4.1.
Reformulation of Gross’ Conjecture
To prove the lemma we use the following two results, the first is from Lemma 1 of [3], and the second is proved using [11]. To deal with the second complex (4.9), we use the following lemma, taken from the Burns–Flach calculations in [2]. The last thing we need to translate Gross's conjecture is a map connecting KΞ to the determinant of the complex RΓc(OK,S, Vp(B)).
The result will follow immediately from the triangle (4.7) if we can show that there exists a natural K⊗QQp-linear isomorphism. The first statement of the Gross conjecture is just the inclusion (4.29), and the p-part of the second is exactly. To study an elliptic curve E with a non-maximal endomorphic ring and its Weil limit B, we use another elliptic curve that is isogenic to E and defined over the same fundamental field F as E .
The morphismυ2 is obtained by observing that the kernel of the morphism. is contained in the nucleus υ1. E0 is the image of a composite morphism:. f α]Ef, which has a foreimage. which is contained in the kernel of the diagonal morphism below:. Therefore, it is the kernel of the above diagonal morphism. which we were originally interested in) is contained in the kernel of υ2 and υ3 can be constructed so that the diagram becomes commutated.
We check that this action is a well-defined ring action and matches the action Oupon E. As before, we have B the Weil limit of E, and we define B0 as the Weil limit of E0.
The Burns–Flach Conjecture
A refinement in the case F (E tors )/K is abelian
In the special case where E is isogenic to a basic change from K to F (as in the examples we construct later), R'K[G].) We have. Taking the rates from Rp to Op (which is possible by Lemma 5.2.1) gives the original Burns–Flach conjecture. In this chapter we present a conjecture in Kato's terminology that is equivalent to the refined Burns–Flach conjecture for B.
The modules involved in the refined Burns-Flach conjecture are in fact isomorphic to the modules just defined above. We use the complex RΓfc(OK,S, T) which in this case is defined by the exact triangle. The vertical left arrow is obtained from dualities and defining exact triangles, so that the composition of the left vertical map and the bottom horizontal map detRp(exp∗⊗id) is another exponential map, which at v |pis exp∗.
Knowing that the last two results are true, we can now state the refined Burns–Flach conjecture in Kato's terminology. The proof will be given in this chapter and more details in the following chapters. Write Km for the beam class field modulo m and let f be the conductor of B. Recall that S is the set of K sites that divides ∞fp. fL is a canonical morphism SpecOL,S →SpecOK,S), F∞ :=lim←−.
Now, our E0/K is an elliptic curve isogenic to our E from Chapter 5 and Chapter 6, and with complex multiplication by the full ring of integers OK.
The strategy of proof
The third line gives the integral submodule of the vector space just above. In the fourth row is a base for the integral submodule (on the left) and its images under the two maps of the second row. The element c+[χ] is defined to be the sub∞ preimage of the element of R with [χ]-component 1 and all other components 0.
HereyΛ∞(OK,S,F∞) is an inverse system of embeddings. which could for example come from a fixed embedding Q,→C. The primary result we will use is. The following lemma relates the cohomology of OK,S to Rubin's Iwasawa modules, to which we will then be able to apply the main conjecture. We wish to consider the [χ] components of R individually, where [χ] is the circuit of χ ∈ Gˆ under AutKC.
Define OK,p-basis ξ of H0(Kab,S, T) to be the image of g−1h under the isomorphisms. where the right isomorphism is induced by ι). In this chapter we will find examples, or show the existence of examples, of elliptic curves for which our main theorem applies. We search for elliptic curves E/F that have complex multiplication with a non-maximal order O = Z+f ·OK for which the Mordell-Weil group E(F) has rank 0.
ZfPf, whereIf is the group of fractional ideals starting up to f, Zf is the group of principal ideals with a generator in Q ⊂K, and Pf is the group of principal ideals with a generator ≡ 1 mod f. Then we will try to show the existence of elliptic curves E/F, isogenic over F, with complex multiplication by O=Z+f ·OK.
Elliptic curves with complex multiplication by a nonmaximal
Our strategy will be to find ring class fields Hf and elliptic curves E0 over F :=Hf with. Moreover, this will show the existence of rank-0 elliptic curves E with non-maximal endomorphism ring, for which only the more general Burns-Flach conjecture applies. We will actually use curves E0 which are defined over K. As before, write ϕ for a Gr¨ossen character of K as when it is precomposed. with the norm, gives the associated Gr¨ossen character ψ of E0 overHf. then we need to verify exactly that.
We know, by the property of the associated Gr¨ossen character, that for x≡1 mod c0, where c0 is the conductor of E0/K. Since OK is by assumption a principle ideal domain, then ϕ is essentially the identity function on K×. Gf is a group of order. where w = |O×K|, φ is Euler's φ-function, and Φ is its obvious analogue defined on ideals of OK.
We observe that 1+f·OK intersects the unit circle at 1 point only if f > 2, which is true here by assumption.
A first attempt
Over the ideals of Z, χ is trivial (since all the ideals of Z represent the trivial class in the ring class group). For each prime number between 3 and 250, the square root W(ϕχ) was numerically checked for all characters χ of Gf. We will determine the first two and show that the fourth can be chosen depending on the third, making the entire product -1.
Now since xx∈Z for x∈OK, and since the image of Z in Gf is trivial, then. The work of Fr¨ohlich and Queyrut ([7]) tells us that for a real character coming from a real representation, the root number is 1. Since the L-function is unchanged by induction or character restriction, we see that W ( χ) must be 1.
A second search
The E0/H3 curve turned out to have a nonzero L value, and thus E3 was the only sample taken from this search. If we knew that L(φ0χ,1)6= 0 for allχ∈Gcf, where φ0 is the Gr¨ossen character of E4,f, then we would be guaranteed the existence of an elliptic curve E4,f defined over Hf , with complex multiplication. exactly with Z+f·OK. The values of L(φ0χ,1) were checked for all characters of the ring class χ, for all prime numbers f between 3 and 61, excluding 17.
All values of L(ϕ0χ,1) were found to be non-zero unless f = 5 and χ was a non-trivial character of the two-element group G5.
The code
Evaluates the character ϕ ='phi' (Gr¨ossen character factor) at 'ideal', given by a generator, expressed as a column vector over the integral basis. We know that the character is either [1] or [3] as a character of the cyclic 4-group 'bnr2.clgp' and, twisting it three times if necessary, we can. Tests whether 'upsilon', a character of the ray class group modulof ('bnr.clgp'), is factorable through the ring class group modulo f, and thus is a valid χ to consider .
Uses the constant 'prim' which is a generator of (Z/fZ)∗ , expressed as an element of 'bnr.clgp'. Tests for 'upsilon' factors through the quotient with (Z/fZ)∗ , i.e. whether it grades the ring class group as well as the ray class group. Of course, varying x does not change the infinite sum, and this was used as a way to check the correctness of the calculation.
Note that in all cases the modulus f of the ring class group is cyclic, but here it is more convenient to consider the characters of the ray class group, finding the value of L only for those factored through the ring class group. Prints "GOOD" if Ef is an elliptic curve of the desired type, and "BAD" otherwise. Checks the L-value for fixedf and all possibleχ, for the beam class group modulus with 2 cyclic components.
Note that for all f ≡ 3 mod 4 f is inert, implying that the beam class group is cyclic modulo f. For all f > 5,≡1 mod 4, f is non-inert and the beam class group has two cyclic components.
