About

Welcome Chao's Math 8674 Topics in number theory!

Time and place: MW 4:10pm-5:25pm in Room 507. First meeting on Sep. 10.

Our topic is Arithmetic of L-functions. More specifically, we will discuss the conjecture of Birch and Swinnerton-Dyer, which predicts deep connections between the L-function of an elliptic curve and its arithmetic, and the vast conjectural generalizations for motives due to Beilinson, Bloch and Kato. In the first half, we will provide necessary background and explain a proof of the BSD conjecture in the rank 0 or 1 case. We will emphasize new tools which generalize to higher dimensional motives. In the second half, we will study recent results on the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives.

Prerequisite: first graduate courses in algebraic number theory, modular forms, and algebraic geometry.

Some tentative topics include:

  • L-functions of elliptic curves and modular forms
  • BSD and BBK conjectures
  • The Waldspurger formula and the Gross-Zagier formula
  • Geometry of modular and Shimura curves
  • Level raising congruences for modular forms
  • Rapoport-Zink weight spectral sequences
  • Bertolini-Darmon explicit reciprocity laws
  • Kolyvagin's method
  • Rankin-Selberg L-functions and Rankin-Selberg motives
  • Gan-Gross-Prasad conjectures and Zhang's formula
  • Geometry of unitary Shimura varieties
  • Tate's conjecture via the geometric Satake correspondence
  • Explicit reciprocity laws for unitary Shimura varieties

Schedule

Lecture Topic Reference
09/10/2018 Overview and the statement of main theorems
09/12/2018 L-functions of elliptic curves and modular forms [S1] V, [S2] I.11, [DS] 5.9
09/17/2018 Rank part of BSD conjecture, computing the leading term, periods [BSD], [Cre] 2.13, [S1] III.5
09/19/2018 Rationality of L-values, heights, full BSD conjecture [Cre] 2.1, [S1] VIII.9, [Gro] 2.4, [Coa] 5
09/24/2018 Tate-Shafarevich groups, Tamagawa number conjecture [S1] X.4, X.6, [Blo]
09/26/2018 Bloch's reformulation of BSD, Selmer groups [Blo], [Zha13], [S1] X.4, [BK]
10/08/2018 Bloch-Kato reformulation, Tate's local duality [BK], [Tat], [Mil], [PR]
10/10/2018 Kolyvagin's method (simplest case), Heegner points [Gro2], [Zha13]
10/15/2018 the Gross-Zagier formula and consequences [GZ], [Zha13]
10/17/2018 the Yuan-Zhang-Zhang formula, the Waldspurger formula [Zha13], [Gro3]
10/22/2018 Weight 3/2 modular forms, Level raising congruences [Gro3], [Rib]
10/24/2018 Ihara's lemma, geometry of modular curves [DT], [LHL] 5.5, [DS] 8
10/29/2018 Geometry of Shimura curves [BD] 5
10/31/2018 Weight spectral sequences [Ill], [Sai], [Liu]
11/07/2018 Examples of weight spectral sequences [Liu]
11/12/2018 First explicit reciprocity law, proof of BSD/BK in rank 0 [BD] 8, 3, 4
11/14/2018 Second explicit reciprocity law, proof of BSD/BK in rank 1 [BD] 9, 3, 4
11/19/2018 Rankin-Selberg L-functions [JPSS], [Cog]
11/26/2018 Gan-Gross-Prasad conjectures, the Ichino-Ikeda-Zhang formula [GGP], [Zha14], [Zha17]
11/28/2018 Bloch-Kato Selmer groups, Bloch-Kato conjecture [Bel]
12/03/2018 BBK for Rankin-Selberg motives, unitary Shimura varieties, GGP cycles [CH], [Shi], [Bei], [Nek]
12/05/2018 Geometry of unitary Shimura varieties, Tate conjecture for the special fiber, explicit reciprocity laws for GGP cycles [VW], [XZ]
12/10/2018 Tate conjecture via geometric Satake [XZ], [Zhu]

References

  • [BD] M. Bertolini, H. Darmon, Iwasawa's Main Conjecture for elliptic curves over anticyclotomic Zp-extensions, Ann. of Math. (2) 162 (2005), no. 1, 1-64.
  • [LTXZZ] Y. Liu, Y. Tian, L. Xiao, W. Zhang, X. Zhu, On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives, in preparation.
  • A plan for MCM summer school organized by Ye Tian, L. Xiao and myself.
  • [S1] J. Silverman, The arithmetic of elliptic curves. Second edition. Graduate Texts in Mathematics, 106.
  • [S2] J. Silverman, Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151.
  • [DS] F. Diamond, J. Shurman, A first course in modular forms. Graduate Texts in Mathematics, 228.
  • [BSD] B. Birch; P. Swinnerton-Dyer, Notes on elliptic curves. II. J. Reine Angew. Math. 218 1965 79-108.
  • [Gro] B. Gross, Lectures on the conjecture of Birch and Swinnerton-Dyer. Arithmetic of L-functions, 169-209, IAS/Park City Math. Ser., 18
  • [Coa] J. Coates, Lectures on the Birch-Swinnerton-Dyer conjecture. ICCM Not. 1 (2013), no. 2, 29-46.
  • [Cre] J. Cremona, Algorithms for modular elliptic curves. Second edition. Cambridge University Press, Cambridge, 1997.
  • [Blo] S. Bloch. A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture. Invent. Math. 58 (1980), no. 1, 65-76.
  • [BK] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, 333-400, Progr. Math., 86, Birkhauser Boston, Boston, MA, 1990.
  • [Zha13] W. Zhang, The Birch-Swinnerton-Dyer conjecture and Heegner points: a survey. Current developments in mathematics 2013, 169-203, Int. Press, Somerville, MA, 2014.
  • [Tat], J. Tate, Duality theorems in Galois cohomology over number fields. 1963 Proc. Internat. Congr. Mathematicians (Stockholm, 1962) pp. 288-295 Inst. Mittag-Leffler, Djursholm.
  • [Mil] J. S. Milne. Arithmetic duality theorems. Second edition. BookSurge, LLC, Charleston, SC, 2006. viii+339 pp.
  • [PR] B. Poonen, E. Rains, Random maximal isotropic subspaces and Selmer groups. J. Amer. Math. Soc. 25 (2012), no. 1, 245-269.
  • [Gro2] Gross, Benedict H. Heegner points on X0(N). Modular forms (Durham, 1983), 87-105, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984.
  • [GZ] Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225-320.
  • [Gro3] Gross, Benedict H. Heights and the special values of L-series. Number theory (Montreal, Que., 1985), 115-187, CMS Conf. Proc., 7, Amer. Math. Soc., Providence, RI, 1987.
  • [Rib] Ribet, Kenneth A, Raising the levels of modular representations. Seminaire de Theorie des Nombres, Paris 1987-88, 259-271, Progr. Math., 81, Birkhauser Boston, Boston, MA, 1990.
  • [DT] Diamond, Fred; Taylor, Richard Lifting modular mod l representations. Duke Math. J. 74 (1994), no. 2, 253-269.
  • [LHL] Le Hung, Bao V.; Li, Chao Level raising mod 2 and arbitrary 2-Selmer ranks. Compos. Math. 152 (2016), no. 8, 1576-1608.
  • [Ill] Illusie, Luc, Nearby cycles and monodromy in etale cohomology, http://176.58.104.245/NOTES/Dobbiaco-2014-06/Illusie-Dobbiaco.pdf
  • [Sai], Saito, Takeshi Weight spectral sequences and independence of l. J. Inst. Math. Jussieu 2 (2003), no. 4, 583-634.
  • [Liu] Yifeng Liu, Bounding cubic-triple product Selmer groups of elliptic curves, Journal of the European Mathematical Society, to appear
  • [JPSS] Jacquet, H.; Piatetskii-Shapiro, I. I.; Shalika, J. A. Rankin-Selberg convolutions. Amer. J. Math. 105 (1983), no. 2, 367-464.
  • [Cog] Cogdell, J. W. Notes on L-functions for GL(n). School on Automorphic Forms on GL(n), 75-158, ICTP Lect. Notes, 21, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008.
  • [GGP] Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I. Asterisque No. 346 (2012), 1-109.
  • [Zha14] Zhang, Wei Automorphic period and the central value of Rankin-Selberg L-function. J. Amer. Math. Soc. 27 (2014), no. 2, 541-612.
  • [Zha17] Zhang, Wei Periods, cycles, and L-functions: a relative trace formula approach, arXiv:1712.08844
  • [Bel] Joel Bellaiche, An introduction to Bloch and Kato's conjecture, http://www.claymath.org/sites/default/files/bellaiche.pdf
  • [CH] Chenevier, Gaetan; Harris, Michael Construction of automorphic Galois representations, II. Camb. J. Math. 1 (2013), no. 1, 573.
  • [Shi] Sug Woo Shin, Construction of Galois representations, https://math.berkeley.edu/~swshin/ClayNotes.pdf
  • [Bei] Beilinson, A. A. Height pairing between algebraic cycles. K-theory, arithmetic and geometry (Moscow, 1984-1986), 1-25, Lecture Notes in Math., 1289, Springer, Berlin, 1987.
  • [Nek] Jan Nekovar, Beilinson's conjectures, Motives (Seattle, WA, 1991), 537-570, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
  • [VW] Vollaard, Inken; Wedhorn, Torsten The supersingular locus of the Shimura variety of GU(1,n-1) II. Invent. Math. 184 (2011), no. 3, 591-627.
  • [XZ] Liang Xiao, Xinwen Zhu, Cycles on Shimura varieties via geometric Satake, arXiv:1707.05700
  • [Z] Xinwen Zhu, Geometric Satake, categorical traces, and arithmetic of Shimura varieties, arXiv:1810.07375