Amod's Mathematics Research

Some Information

• CV:    pdf.
  

Papers/preprints

• Computing intersection numbers between abelian varieties associated to subspaces of modular forms, submitted. pdf.
  
• On the notion of visibility of torsors, published. pdf.
  
• The special L-value of the winding quotient of level a product of two distinct primes, submitted. pdf.
  
• Conjectures concerning the orders of the torsion subgroup, the arithmetic component groups, and the cuspidal subgroup, submitted. pdf.
  
• Appendix on periods, in: V. Pal, Periods of quadratic twists of elliptic curves, published. pdf.
  
• Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero, submitted. pdf.
  
• Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group, J. Ramanujan Math. Soc. 26, No. 4 (2011), 357--367. pdf.
  
• The modular number, the congruence number, and multiplicity one, submitted pdf.
  
• A visible factor of the Heegner index, Math. Res. Lett. 17 (2010), no. 6, 1065--1077. pdf.
  
• Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one, Int. Math. Res. Not. (IMRN), Vol. 2009 (2009), No. 15, pp. 2899--2913 pdf.
  
• Squareness in the special L-value and special L-values of twists, Int. J. Number Theory, Vol. 6, No. 5 (2010), 1091--1111 pdf.
  
• Rational torsion in elliptic curves and the cuspidal subgroup, submitted. pdf.
  
• The modular degree, congruence primes, and multiplicity one (with K. Ribet and W. Stein), in "Number Theory, Analysis and Geometry, In Memory of Serge Lang", Springer, (2012), 19--50, pdf.
  
• A visible factor of the special L-value, J. Reine Angew. Math. (Crelle's journal), Vol. 644 (2010), 159--187.    pdf.
  
• The Manin constant (with K. Ribet and W. Stein), Pure and Applied Mathematics Quarterly (special issue in honor of John Coates), 2 (2006), no.~2, 617--636 : pdf.
  
• The Manin constant, congruence primes, and the modular degree (with K. Ribet and W. Stein), unpublished older version: ps, pdf.
  
• Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero (with W. Stein, and with an appendix by J. Cremona and B. Mazur) Mathematics of Computation 74 (2005), no.~249, 455--484. ps, pdf.
  
• Constructing elliptic curves with known number of points over a prime field (with K. Lauter and R. Venkatesan), in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Institute Communications, vol.~41, Amer. Math. Soc., Providence, RI, 2004, pp.~1--17. ps, pdf.
  
• Visibility of Shafarevich-Tate Groups of Abelian Varieties (with W.Stein), J. Number Theory 97 (2002), no. 1, 171--185.    ps, pdf.
  
• Appendix on generating the Hecke algebra (with W. Stein), in: J.-C. Lario, R. Schoof, Some computations with Hecke rings and deformation rings, Experiment. Math. 11 (2002), no. 2, 303--311. ps, pdf.
  An earlier non-appendix version: ps, pdf.
  
• On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374. Abridged version:    ps, pdf.
  
• Ph.D. Thesis (Warning: There are some minor mistakes in Chapter 2: some of the results should be stated only for newform quotients; also the proof of the result regarding the relation between the Manin constant, congruence primes and the modular degree has a gap. Please refer to the Manin constant article above for the correct statements). dvi     pdf    

Informal, Expository, etc.:

• In progress: a book titled "A motivated introduction to algebraic geometry"   link.
  
• The Birch and Swinnerton-Dyer conjecture (a proposal for a local grant that is accessible to people in areas related to mathematics)   pdf.
  
• Rational torsion points on elliptic curves (a poster for an informal workshop, again accessible to people in areas related to mathematics)   pdf.
  
• Coming soon: "A motivated introduction to class field theory"
  
• Coming sometime in the future: "A motivated introduction to stacks"
  

Slides from some talks

• AMS national meeting, San Francisco, 2010   pdf. (needs some corrections)
  
• AMS Sectional meeting, Boca Raton   pdf. (Note: there are some missing figures)
  
• AMS Sectional meeting, Fayetteville   pdf. (Note: there are some missing figures)
  
• AMS National meeting, San Antonio, and AMS Sectional meeting, San Francisco   ps, pdf. (Note: there is some repetition in the slides, and some figures are missing)
  
• Barcelona conference on "Modular curves and abelian varieties"      ps, pdf.
  
• AMS Sectional meeting, Lawrence, Kansas      ps, pdf.
  
For some tables, see W. Stein's web pages:      tables1     tables2

• AMS National meeting, New Orleans      ps, pdf.
  
• Millennial conference in Urbana      ps. pdf.