TITLE: Torsion points on elliptic curves over quartic number fields SPEAKER: William Stein ABSTRACT: Loic Merel proved that for every positive integer d, there is a bound B_d such that if E is an elliptic curve over a number field K of degree d, then #E(K)_tor <= B_d. A natural question is then to classify all possibilities for E(K)_tor, as E/K varies over all elliptic curves over degree d number fields. When d=1, Barry Mazur answered this question, in his famous 1977 paper "Modular Curves and the Eisenstein Ideal". When d=2 and d=3 this question has also been answered, due to work of Abromovich, Kamienny, Parent and others. This talk will be mainly about the still unresolved case d=4, toward which Sheldon Kamienny and I have made recent progress. ----------------------------------------- PROBLEM: Let K be a number field. Which finite abelian groups E(K)_{tor} occur, as we vary over all elliptic curves E/K? There are a *LOT* of papers on this problem. OBSERVATION: E(K)_{tor} is a finite subgroup of Q^2/Z^2, so E(K)_{tor} is cyclic or a product of two cyclic groups. ----------------------------------------- Elliptic curves over QQ CONJECTURE (LEVI around 1908; OGG in 1960s): When K=Q, the groups E(Q)_{tor} are the 15 groups: Z/mZ for m<=10 or m=12 (Z/2Z) x (Z/2vZ) for v<=4. NOTE: Moreover, each of these groups occurs for infinitely many j-invariants. THEOREM (Kubert): Suppose that we know that whenever a prime p divides #E(Q)_tor, then p<=7. Then Ogg's conjecture is true. Modular Curves and the Eisenstein Ideal THEOREM (Mazur, 1973): If p | #E(Q)_tor, then p <= 13. Idea of Proof: ... ----------------------------------------- The Method of Kamienny-Mazur DEFINITION: A prime p is a TORSION PRIME for degree d if there is a number field K of degree d and an elliptic curve E/K such that p divides #E(K)_tor. S(d) = { set of torsion primes of degree <= d }. PROBLEM: Determine the torsion primes for each degree. EXAMPLE: Mazur's theorem above implies that S(1) = {2,3,5,7} THEOREM (Kamienny and Mazur, 1977-1992): S(1) = {2,3,5,7} S(2) = {2,3,5,7,11,13} S(d) is finite for d <= 8, and S(d) is of density 0 for all d. Idea of Proof: ... CONJECTURE (Kamienny): S(d) is finite for all d. This conjecture implies that the set of groups E(K)_tor is finite, as E varies over all elliptic curves over all number fields of degree d. THEOREM (Abromovich) S(d) is finite for d=9,10,11,12,13,14. THEOREM (Kenku-Momose and Kamienny-Mazur, 1992): If E is an elliptic curve over a quadratic field K, the E(K)_tor is one of the following groups Z/mZ for m<=16 or m=18 (Z/2Z) x (Z/2vZ) for v<=6. (Z/3Z) x (Z/3vZ) for v=1,2 (Z/4Z) x (Z/4ZZ) ----------------------------------------- Merel's Uniform Boundeness Theorem THEOREM (Merel, 1996): max(S(d)) < d^(3*d^2), so S(d) is finite for all d. REMARK: Oesterle improved the bound: max(S(d)) < (3^(d/2)+1)^2 d | max(S(d)) bound --------------------- 1 | 7 2 | 16 3 | 38 4 | 100 5 | 275 6 | 784 COROLLARY: Let d>=1 be an integer. The set of torsion subgroups E(K)_{tor} of elliptic curves E over number fields K of degree <= d is finite. Idea of proof... [see/summarize Rebolledo] The key idea is to replace the Eisenstein quotient of Kamienny-Mazur by the winding quotient, apply the BSD conjecture to get that it has rank 0 (which is known here due to Gross-Zagier & Kolyvagin-Logachev), and then verifies Kamienny's criterion for N>>0 using modular symbols and the intersection pairing. [[See Darmon's MSN review of this paper for a great overview.]] ----------------------------------------- The Method of Jeon, Kim et al. Observations: * Over QQ, each of the torsion subgroup structures that shows up appears for infinitely many j-invariants. * THEOREM (3.5 in Jeon-Kim_park): Over quadratic fields K, each of the torsion subgroup structures that shows up appears for infinitely many j-invariants. THEOREM (Jeon, Kim, Schweizer, 2004): As K varies over cubic fields, the groups E(K)_tor that occur for infinitely many j-invariants are: Z/mZ for m<=16, 18, 20 Z/2Z x Z/2vZ for v<=7 [[Are these everything? Parent must say, right?]] THEOREM (Jeon, Kim, Park, 2006): As K varies over quartic fields, the groups E(K)_tor that occur for infinitely many j-invariants are: Z/mZ for m<=18, or m=20, m=21, m=22, m=24 Z/2Z x Z/2vZ for v<=9 Z/3Z x Z/3vZ for v<=3 Z/4Z x Z/4vZ for v<=2 Z/5Z x Z/5Z Z/6Z x Z/6Z Idea of proof: ... ----------------------------------------- QUESTION: Which torsion subgroups appear over quartic fields? There could a priori be many other groups not listed above that appear only finitely many times. The method of Kamienny-Parent: THEOREM (Parent): S(3) = {2,3,5,7,11,13} THEOREM (Kamienny, Stein, 2010): We have max(S(4)) <= 31. Idea of Proof: ... I show that max(S(4)) <= 31 using Kamienny criterion as in Parent and specialized to degree 4. If we tried to use ell=3, the criterion isn't too ridiculously hard to check. But then we can only get results about p > (1+3^(d/2))^2 = (1+3^2)^2 = 100! Argh!! So we have to consider the formal immersion criterion with ell=2, and for that one must work directly with X_1(N), as in [Parent, 1999]. Also, we have to use Kato instead of Kolyvagin to deduce that the J_1 version of the winding quotient has rank 0 ( Note: Kamienny I think has somehow shown 29, 31 are also not in S(4). (??) ----------------------------------------- Directions for future work: * Explicitly study the modular curves X_1(N) for N=19 and 23 (genus 7 and 12) to decide if these values occur as the order of quartic torsion points. This would finish the computation of S(4). Note that 19 and 23 are dealt with for many sorts of quartic fields by unpublished worked of Kamienny. * Make the algorithm for bounding S(d) much more efficient and run for d=5.