Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication.

*(English)*Zbl 0628.14018Let K be a number field and \(E/K\) an elliptic curve. The conjecture of Birch and Swinnerton-Dyer gives a relation between the behavior of the L- series \(L(E_{/K},s)\) around \(s=1\) and (among other quantities) the order of the Tate-Shafarevich group Russian{Sh} = Russian{Sh}\((E_{/K})\). As described by J. T. Tate [Invent. Math. 23, 179-206 (1974; Zbl 0296.14018)], “this remarkable conjecture relates the behavior of a \(function\quad L\) at a point where it is not at present known to be defined to the order of a group {Russian{Sh}} which is not known to be finite!” In this important paper the author gives the first examples of elliptic curves for which it can be proved that the Tate- Shafarevich group is finite. His first result gives a relation between the value of \(L(E_{/K},1)\) and the order of {Russian{Sh}}; and his second relates the order of vanishing of \(L(E_{/K},s)\) at \(s=1\) to the rank of the Mordell-Weil group E(K). Both of these results provide additional evidence for the truth of the Birch and Swinnerton-Dyer conjecture. We now describe the author’s results in more detail.

Let \(E/K\) be an elliptic curve with complex multiplication by an order \({\mathfrak O}\) in the imaginary quadratic field K, let \({\mathfrak O}_ K\) be the ring of integers of K, and let \(\Omega\) be an \({\mathfrak O}\)-generator of the period lattice of a minimal model for E. Let \(\psi\) be the Hecke character of K attached to E. The L-function of \(E/K\) satisfies \(L(E_{/K},s)=L(\psi,s)L({\bar \psi},s)\), and \(L({\bar\psi},1)/\Omega\in K.\)

Theorem A. (a) If \(L(E_{/K},1)\neq 0\), then {Russian{Sh}} is finite. (b) Let \({\mathfrak p}\) be a prime of K not dividing \(| {\mathfrak O}^*_ K|\). If \(| E(K)_{tors}| L({\bar\psi},1)/\Omega \not\equiv 0\) (mod \({\mathfrak p})\), then the \({\mathfrak p}\)-part of {Russian{Sh}} is trivial.

Theorem B. Let E be an elliptic curve defined over \({\mathbb{Q}}\) with complex multiplication. If \(\text{rank}_{{\mathbb{Z}}}(E({\mathbb{Q}}))\geq 2\), then \(\text{ord}_{s=1}(E_{/{\mathbb{Q}}},s)\geq 2.\)

The author’s proofs rely heavily on the techniques originally developed by J. Coates and A. Wiles [Invent. Math. 39, 223-251 (1977; Zbl 0359.14009) and J. Aust. Math. Soc., Ser. A 26, 1-25 (1978; Zbl 0442.12007)], in particular on a refinement of the relation between elliptic units and L(\({\bar \psi}\),1). The main new ingredient is the use of ideal class annihilators arising from elliptic units, which was suggested by the work of Thaine on cyclotomic units and class groups of cyclotomic fields [“On the ideal class groups of real abelian number fields ” (to appear)]. This allows the author to control the size of a certain class group while working entirely in the field \(K(E_{{\mathfrak p}})\); the original work of Coates and Wiles (op. cit.) required using \(K(E_{{\mathfrak p}^ n})\) for all \(n\geq 1.\)

As the author indicates in his introduction, the major complications in the proof of theorem A arise from a small number of primes, in particular primes of bad reduction and primes dividing \(| {\mathfrak O}^*_ K|\). The interested reader might start by reading the proof of the weaker (but still striking) statement “If \(L(E_{/K},1)3n0\) then {Russian{Sh}} has no \({\mathfrak p}\)-torsion for almost all \({\mathfrak p}.''\) The technical details needed to complete the proof of theorem A can then be found in the later sections.

The proof of theorem B is essentially independent from that of theorem A, although it again relies heavily on the use of elliptic unit ideal class annihilators. It also uses recent works of B. H. Gross and D. Zagier [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] and B. Perrin-Riou [“Points de Heegner et dérivés de fonctions L p-adic,” Invent. Math. (to appear)]. As a corollary of (the proof of) theorem B and the Gross-Zagier theorem (op. cit.), the author also deduces that if E is defined over \({\mathbb{Q}}\) and has complex multiplication, and if \(L(E_{/{\mathbb{Q}}},s)\) has a simple zero at s-1, then the p-part of {Russian{Sh}}\((E_{/{\mathbb{Q}}})\) is finite for all primes p.2 for which E has good, ordinary reduction (i.e. for approximately half the primes p).

Let \(E/K\) be an elliptic curve with complex multiplication by an order \({\mathfrak O}\) in the imaginary quadratic field K, let \({\mathfrak O}_ K\) be the ring of integers of K, and let \(\Omega\) be an \({\mathfrak O}\)-generator of the period lattice of a minimal model for E. Let \(\psi\) be the Hecke character of K attached to E. The L-function of \(E/K\) satisfies \(L(E_{/K},s)=L(\psi,s)L({\bar \psi},s)\), and \(L({\bar\psi},1)/\Omega\in K.\)

Theorem A. (a) If \(L(E_{/K},1)\neq 0\), then {Russian{Sh}} is finite. (b) Let \({\mathfrak p}\) be a prime of K not dividing \(| {\mathfrak O}^*_ K|\). If \(| E(K)_{tors}| L({\bar\psi},1)/\Omega \not\equiv 0\) (mod \({\mathfrak p})\), then the \({\mathfrak p}\)-part of {Russian{Sh}} is trivial.

Theorem B. Let E be an elliptic curve defined over \({\mathbb{Q}}\) with complex multiplication. If \(\text{rank}_{{\mathbb{Z}}}(E({\mathbb{Q}}))\geq 2\), then \(\text{ord}_{s=1}(E_{/{\mathbb{Q}}},s)\geq 2.\)

The author’s proofs rely heavily on the techniques originally developed by J. Coates and A. Wiles [Invent. Math. 39, 223-251 (1977; Zbl 0359.14009) and J. Aust. Math. Soc., Ser. A 26, 1-25 (1978; Zbl 0442.12007)], in particular on a refinement of the relation between elliptic units and L(\({\bar \psi}\),1). The main new ingredient is the use of ideal class annihilators arising from elliptic units, which was suggested by the work of Thaine on cyclotomic units and class groups of cyclotomic fields [“On the ideal class groups of real abelian number fields ” (to appear)]. This allows the author to control the size of a certain class group while working entirely in the field \(K(E_{{\mathfrak p}})\); the original work of Coates and Wiles (op. cit.) required using \(K(E_{{\mathfrak p}^ n})\) for all \(n\geq 1.\)

As the author indicates in his introduction, the major complications in the proof of theorem A arise from a small number of primes, in particular primes of bad reduction and primes dividing \(| {\mathfrak O}^*_ K|\). The interested reader might start by reading the proof of the weaker (but still striking) statement “If \(L(E_{/K},1)3n0\) then {Russian{Sh}} has no \({\mathfrak p}\)-torsion for almost all \({\mathfrak p}.''\) The technical details needed to complete the proof of theorem A can then be found in the later sections.

The proof of theorem B is essentially independent from that of theorem A, although it again relies heavily on the use of elliptic unit ideal class annihilators. It also uses recent works of B. H. Gross and D. Zagier [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] and B. Perrin-Riou [“Points de Heegner et dérivés de fonctions L p-adic,” Invent. Math. (to appear)]. As a corollary of (the proof of) theorem B and the Gross-Zagier theorem (op. cit.), the author also deduces that if E is defined over \({\mathbb{Q}}\) and has complex multiplication, and if \(L(E_{/{\mathbb{Q}}},s)\) has a simple zero at s-1, then the p-part of {Russian{Sh}}\((E_{/{\mathbb{Q}}})\) is finite for all primes p.2 for which E has good, ordinary reduction (i.e. for approximately half the primes p).

Reviewer: J.H.Silverman

##### MSC:

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14K22 | Complex multiplication and abelian varieties |

14H45 | Special algebraic curves and curves of low genus |

14G05 | Rational points |

14H52 | Elliptic curves |

##### Keywords:

finite Tate-Shafarevich group; elliptic curves; evidence for the truth of the Birch and Swinnerton-Dyer conjecture; ideal class annihilators##### References:

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