Henri Darmon

Professor,
Department of Mathematics.

Office: Burnside Hall, Room 1221
Office Hours: By request.
Office Phone: (514) 398-2263
Office Fax: (514) 398-3899

Research Interests

Algebraic number theory, with a special emphasis on elliptic curves, modular forms, and their associated L-functions.

Elliptic Curves and the Birch and Swinnerton-Dyer conjecture

My Harvard PhD thesis (1991; cf. also [Da1]) formulated an analogue of a conjecture of Mazur and Tate [Mt1], replacing cyclotomic extensions of Q by anticyclotomic extensions of an imaginary quadratic field and modular symbols with Heegner points. One could apply the ideas of Kolyvagin [Ko1], which had just appeared then, to prove special cases of this conjecture which could not be approached in the original setting considered by Mazur and Tate.

One of the puzzling features of the anticyclotomic setting is the presence of predictable degeneracies in the associated height pairings. (In the cyclotomic case, a conjecture of Mazur predicts that the cyclotomic p-adic height pairing is always non-degenerate.) Such degeneracies lead to extra vanishing of the corresponding p-adic L-functions. The phenomenon of extra vanishing brought about by degeneracies in the height pairings was studied in a series of joint papers ([BD1], [BD2]) with M. Bertolini, where the notion of a "derived height" was introduced.

The conjectures of Mazur and Tate are themselves a refinement of some earlier conjectures of Mazur, Tate and Teitelbaum [MTT] which are p-adic variants of the Birch and Swinnerton Dyer conjecture. The conjectures of Mazur, Tate and Teitelbaum were transposed to the anti-cyclotomic context in [BD3]. Special cases of these conjectures were then proved in [BD4], [BD5], and [BD6]. The proof of the main result of [BD5] exploits the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves to construct Heegner points on elliptic curves from derivatives of p-adic L-functions.

Recently, Bertolini and I [BD7] have given a partial proof of the ``main conjecture" of Iwasawa theory for elliptic curves in the anticyclotomic setting, which implies in particular that the order of vanishing of the anti-cyclotomic $p$-adic $L$-function is at least equal to the rank of the corresponding Mordell-Weil group. (The corresponding result, in the cyclotomic setting, has been proved by Kato.)

Complex Multiplication for real quadratic fields

The desire to understand the role played by p-adic uniformisation in the study of p-adic L-functions in [BD5] and [BD6] and to reconcile the ostensibly different approaches used to treat p-adic L-functions in the cyclotomic and anticyclotomic settings led me to introduce [Da5] the notion of modular forms of weigt (2,2) on H_p x H and to define p-adic periods associated to such forms. This clarifies the relation between the exceptional zero conjectures of Mazur, Tate and Teitelbaum and my study with Bertolini of the anticyclotomic case. Most significantly perhaps, the article [Da5] provides a conjectural p-adic analytic construction of global points on elliptic curves, points which are defined over ring class fields of certain real quadratic fields. This is intriguing, insofar as it suggests that the theory of complex multiplication should extend (at least conjecturally) to real quadratic fields. It is still an open question (known as Hilbert's 12th problem, or Kronecker's Jugendtraum) to supply analytic constructions of the class fields of real quadratic fields (or of more general number fields) along the lines of what is accomplished by the theory of complex multiplication for imaginary quadratic fields.

Fermat's Last Theorem

In June 1993, Andrew Wiles announced his proof of the Shimura Taniyama conjecture, conquering Fermat's Last Theorem in the same stroke. A few months before I had been thinking about Frey's ideas and applying them to study the generalized Fermat equation x^p+y^q=z^r. One is interested in integer solutions which are primitive in the sense that they satisfy gcd(x,y,z)=1, and the generalized Fermat conjecture states that there are no such solutions when the exponents p,q,r are greater than 3. At the time I could prove some partial results [Da2] about x^p+y^p =z^r with r=2 or 3, assuming the Shimura Taniyama conjecture. Thanks to [TW] and [W] these results became unconditional. More importantly, the general program of tackling such ternary diophantine equations via modular forms and "GL(2) reciprocity laws" began to seem more attractive.

By generalizing certain results of Mazur on torsion of elliptic curves, the article [DM] (joint with L. Merel) proved that the equations xⁿ+yⁿ =z^r (r=2 or 3) has no non-trivial primitive solution when the exponent n is greater than 4.

The article [Da3] extends the approach of [Da2] and [DM] to the generalized Fermat equation. In spite of the fundamental new ideas introduced by Wiles, one still encounters substantial difficulties in the study of the generalized Fermat equation, and [Da3] does not yield any new cases of the generalized Fermat conjecture beyond those covered in [DM]! Nonetheless, the generalized Fermat equation seems to share with its classical couterpart the power of generating important mathematical questions. For example, one is naturally led to replace elliptic curves by certain "hypergeometric abelian varieties", so named because their periods are related to values of hypergeometric functions. The methods of Wiles suggest a workable strategy for establishing the modularity of a large class of hypergeometric abelian varieties [Da4].

References

[BD1] Bertolini, Massimo; Darmon, Henri Derived heights and generalized Mazur-Tate regulators. Duke Math. J. 76 (1994), no. 1, 75--111.

[BD2] Bertolini, Massimo; Darmon, Henri Derived $p$-adic heights. Amer. J. Math. 117 (1995), no. 6, 1517--1554.

[BD3] Bertolini, M.; Darmon, H. Heegner points on Mumford-Tate curves. Invent. Math. 126 (1996), no. 3, 413--456.

[BD4] Bertolini, Massimo; Darmon, Henri A rigid analytic Gross-Zagier formula and arithmetic applications. With an appendix by Bas Edixhoven. Ann. of Math. (2) 146 (1997), no. 1, 111--147.

[BD5] Bertolini, Massimo; Darmon, Henri Heegner points, $p$-adic $L$-functions, and the Cerednik-Drinfeld uniformization. Invent. Math. 131 (1998), no. 3, 453--491.

[BD6] Bertolini, Massimo; Darmon, Henri. p-adic periods, p-adic L-functions and the p-adic uniformization of Shimura curves. Duke Math J., 98 (1999) 305-334.

[BD7] Bertolini, Massimo; Darmon, Henri. Iwasawa's main conjecture for elliptic curves over anticyclotomic Z_p-extensions. in progress.

[Da1] Darmon, Henri. A refined conjecture of Mazur-Tate type for Heegner points. Invent. Math. 110 (1992), no. 1, 123--146.

[Da2] Darmon, Henri. The equations xⁿ+yⁿ=z² and xⁿ+yⁿ=z³. Internat. Math. Res. Notices 1993, no. 10, 263--274.

[Da3] Darmon, Henri. Rigid local systems, Hilbert modular forms, and Fermat's Last Theorem. Duke Math j. 102 (2000) 413-449.

[Da4] Darmon, Henri. Modularity of fibers in rigid local systems. Annals of Math, 149 (1999) 1079-1086.

[Da5] Darmon, Henri. Integration on Hp x H and arithmetic applications. Submitted.

[DM] Darmon, Henri; Merel, Loic. Winding quotients and some variants of Fermat's last theorem. J. Reine Angew. Math. 490 (1997), 81--100.

[Ko1] Kolyvagin, V. A. The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1154--1180, 1327; translation in Math. USSR-Izv. 33 (1989), no. 3, 473--499.

[MT1] Mazur, B.; Tate, J. Refined conjectures of the "Birch and Swinnerton-Dyer type". Duke Math. J. 54 (1987), no. 2, 711--750.

[MTT] Mazur, B.; Tate, J.; Teitelbaum, J. On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1--48.

[TW] Taylor, Richard; Wiles, Andrew Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553--572.

[W] Wiles, Andrew Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443--551.