Heegner points and cycles PhD seminar

Since the work of Gross-Zagier and Kolyvagin it is clear that Heegner points on elliptic curves are a poweful arithmetic tool. The aim of this seminar is to give a brief introduction to this nowadays classical theory and to show more recent developments in this direction, such as Heegner points on Shimura curves and Heegner cycles attached to a modular form.

Schedule

1. Heegner points on modular elliptic curves.  Definition of Heegner points on a modular elliptic curves, Kolyvagin's derivative and its application to the study of the structure of Selmer groups. References: [Gro91]. Date: Th 31st March 2022, 15:00-16:30 - Room: 2BC60 Speaker: Marco Baracchini

2. Iwasawa Theory of elliptic curves. Z_p-extensions, Iwasawa algebra, Iwasawa modules, Selmer groups of an elliptic curve over Z_p-extensions.  References: [Was97], [NSW08],  [Gre99], [Ski18], [Ouy]. Date: Th 7th April 2022, 16:30-18:00 - Room: 2AB45. Speaker: Nguyen-Dang Khai-Hoan

3. Iwasawa theory of Heegner points and Kolyvagin system I. Definition of (anticyclotomic) Kolyvagin system and construction of the Kolyvagin system of Heegner points. References: [Per87], [How04b]. Date: Fr 8th April 2022, 10:30-12:00  - Room: 2AB40. Speaker:Luca Mastella

4. Iwasawa theory of Heegner points and Kolyvagin system II. Anticyclotomic extension, Perrin-Riou main conjecture. Date: Th 11th April 2022. Speaker:Luca Mastella

5. Shimura Curves. Definition of quaternion algebras, Shimura curves of PEL type arising from quaternion algebras. Date: Th 21st April 2022, 16:30-18:00 - Room: 2AB45.  Speaker: Daniele Troletti

6. Heegner points on Shimura Curves CM points on Shimura curves, Heegner points and Kolyvagin system.  References: [Nek07], [How04a].  Date: Fr 22nd April 2022, 10:30-12:00 - Room: SR701.  Speaker: Daniele Troletti

7. Heegner cycles I. Kuga-Sato varieties and definition of Heegner cycles. References: [Nek92], [BDP13]. Date: Th. 5th May 2022.  Speaker:Luca Mastella

8. Heegner cycles II. Arithmetic applications, Kolyvagin system of Heegner cycles. References: [CH18], [LV19]. Date: Fr. 6th May 2022 Speaker:Luca Mastella

9. Big Heegner Points: Definition of Big Heegner points for Hida families. References [How07], [ Büy14] , [ LV11].  Date: Th. 12th May 2022. Speaker: Francesco Zerman.

References

[Büy14] Kâzım Büyükboduk,  “Big Heegner point Kolyvagin system for a family of modular forms”. In: Selecta Math. (N.S.) 20 (2014), no.3, pp. 787–815.

[BDP13] Massimo Bertolini, Henri Darmon, and Kartik Prasanna. “Generalized Heegner cycles and p-adic Rankin L-series”. In: Duke Math. J. 162.6 (2013). With an appendix by Brian Conrad, pp. 1033–1148.

[CH18] Francesc Castella and Ming-Lun Hsieh. “Heegner cycles and p-adic L-functions”. In: Math. Ann. 370.1-2 (2018), pp. 567–628.

[Gre99] Ralph Greenberg. “Iwasawa theory for elliptic curves”. In: Arith- metic theory of elliptic curves (Cetraro, 1997). Vol. 1716. Lecture Notes in Math. Springer, Berlin, 1999, pp. 51–144. 

[Gro91] Benedict H. Gross. “Kolyvagin’s work on modular elliptic curves”. In: L-functions and arithmetic (Durham, 1989). Vol. 153. London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 1991, pp. 235–256.

[How04a] Benjamin Howard. “Iwasawa theory of Heegner points on abelian varieties of GL2 type”. In: Duke Math. J. 124.1 (2004), pp. 1–45.

[How04b] Benjamin Howard. “The Heegner point Kolyvagin system”. In: Compos. Math. 140.6 (2004), pp. 1439–1472. 

[How07] Benjamin Howard. “Variation of Heegner points in Hida families”. In: Invent. Math.167(2007), no.1, pp. 91–128.

[LV11]  Matteo Longo and Stefano Vigni. “Quaternion algebras, Heegner points and the arithmetic of Hida families”. In: Manuscripta Math.135(2011), no.3-4, pp. 273–328.

[LV19] Matteo Longo and Stefano Vigni. “Kolyvagin systems and Iwasawa theory of generalized Heegner cycles”. In: Kyoto J. Math. 59.3 (2019), pp. 717–746.

[Nek92] Jan Nekovář. “Kolyvagin’s method for Chow groups of Kuga-Sato varieties”. In: Invent. Math. 107.1 (1992), pp. 99–125.

[Nek07] Jan Nekovář. “The Euler system method for CM points on Shimura curves”. In: L-functions and Galois representations. Vol. 320. London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 2007, pp. 471–547.

[NSW08] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg. Coho- mology of number fields. Second. Vol. 323. Grundlehren der mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2008, pp. xvi+825.

[Ouy] Y. Ouyang. “Introduction to Iwasawa Theory”. 

[Per87] Bernadette Perrin-Riou. “Fonctions L p-adiques, théorie d’Iwasawa et points de Heegner”. In: Bull. Soc. Math. France 115.4 (1987), pp. 399–456. 

[Ski18] C. Skinner. “Iwasawa Theory, modular forms and elliptic curves”. In: Arizona Winter school Lecture Notes. 2018.

[Was97] Lawrence C. Washington. Introduction to cyclotomic fields. Second. Vol. 83. Graduate Texts in Mathematics. Springer-Verlag, New York, 1997, pp. xiv+487.