Amod's Mathematics Research

Some Information

• CV:    pdf.
  

Papers/preprints

• (with M. Winters) Reducibility and rational torsion in elliptic curves, submitted. pdf.
  
• The modular number, the congruence number, and multiplicity one, submitted pdf.
  
• A generalization of Kronecker's first limit formula, submitted. pdf.
  
• (with M. Trifkovic) Darmon points on elliptic curves over totally real fields, J. Ramanujan Math. Soc. 37 (2022), no. 3, 231--239. pdf.
  
• Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero, J. Ramanujan Math. Soc. 36 (2021), no. 2, 169--178. pdf.
  
• Rational torsion in elliptic curves and the cuspidal subgroup, Journal de theorie de nombres de Bordeaux, 30 (2018), no. 1, 81-91. pdf.
  
• (with L. Eldredge) On the cohomology groups of certain quotients of products of upper half planes and upper half spaces, C. R. Acad. Sci. Paris Ser. I Math. 355 (2017), 937--941. Slightly enlarged version:    pdf.
  
• The special L-value of the winding quotient of level a product of two distinct primes, Manuscripta Math. 151 (2016), no. 3-4, 493-504. pdf.
  
• (with R. Heaton) Computing intersection numbers between abelian varieties associated to subspaces of modular forms, J. Symbolic Comput. 57 (2013), 70-77 pdf.
  
• (with S. Biswas) Constructing non-trivial elements of the Shafarevich-Tate group of an abelian variety over a number field. J. Number Theory 133 (2013), no. 6, 1977 -- 1990.
  
• On the notion of visibility of torsors, Canadian Mathematical Bulletin 56(2013), no. 2, 225-228. pdf.
  
• Conjectures concerning the orders of the torsion subgroup, the arithmetic component groups, and the cuspidal subgroup, Exp. Math. 22 (2013), no. 4, 363-366. pdf.
  
• Appendix on periods, in: V. Pal, Periods of quadratic twists of elliptic curves, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1513 -- 1525 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.
  
• Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group, J. Ramanujan Math. Soc. 26, No. 4 (2011), 357--367. pdf.
  
• A visible factor of the Heegner index, Math. Res. Lett. 17 (2010), no. 6, 1065--1077. pdf.
  
• A visible factor of the special L-value, J. Reine Angew. Math. (Crelle's journal), Vol. 644 (2010), 159--187.    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.
  
• 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.
  
• 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.
  

Slides from some talks

• MASON V (online)   pdf.
  
• AMS Sectional meeting, Gainesville, FL   pdf.
  
• AMS Sectional meeting, Athens, GA   pdf.
  
• 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.