Kevin James

Summary of Professional Contributions ( CV , Short CV )

• Research

  I am a member of the Clemson Number Theory Group . I am particularly interested in the arithmetic theory of modular forms and elliptic curves. The following is an outline of some of my more significant publications grouped by area.

  • Ranks of Elliptic Curves and Nonvanishing of L-series

    The following papers established important examples of families of quadratic twists of elliptic curves having the property that a positive proportion of the curves in the family have an $L$-series which is provably nonzero (1) or whose Selmer group is trivial (2). The third paper exploits congruences established in (1) in order to establish the Birch and Swinnerton-Dyer conjecture modulo 3 for certain curves.
    
    
    1. $L$-series with non-zero central critical value, Journal of the American Mathematical Society, 11 (1998), 635--641.
    2. (with Ken Ono) Selmer groups of quadratic twists of elliptic curves, Math. Ann., 314, (1999), no. 1, 1--17.
    3. Elliptic Curves satisfying the Birch and Swinnerton-Dyer conjecture mod 3, Journal of Number Theory, 76 (1999), 16--21.

  • Distribution of Primes and the Arithmetic of Elliptic Curves and Modular Forms

    The first two papers show that the Lang-Trotter conjecture is true on average for families of elliptic curves with certain torsion substructures. The third paper shows that a refinement of the Sato-Tate conjecture is true on average when one averages over all elliptic curves. The latter papers prove that the Lang-Trotter conjecture is true on average for elliptic curves defined over certain number fields.
    
    
    1. Average Frobenius distributions for elliptic curves with 3-torsion, Journal of Number Theory 109 (2004) no. 2, 278--298.
    2. (with REU Students: J. Battista, J. Bayless, D. Ivanov), Average Frobenius distributions for elliptic curves with nontrivial rational torsion, Acta Arith. 119(1) (2005), 81--91.
    3. (with Gang Yu), Average Frobenius distribution of elliptic curves, Acta Arith. 124 (2006), 79-100.
    4. (with N. Calkin, D. Penniston; PhD Student: B. Faulkner and REU Student: M. King), Average Frobenius distributions for elliptic curves over abelian extensions, Acta Arith. 149 (2011), no. 3, 215-244.
    5. (with PhD. Student: E. Smith), Average Frobenius Distributions for elliptic curves over Galois extensions, Math Proc Camb Phil Soc. 150 issue 03 (2011) 439--458.
    6. (with PhD Student: E. Smith), Average Frobenius distribution for the degree two primes of a number field, (accepted modulo revision, revised and resubmitted).

  • Simplicity of the Hecke Algebra for Level 1 Modular Forms and Factorization of Eigenforms

    The first two papers establish results which give evidence for Maeda's conjecture on the simplicity of the Hecke Algebra for modular forms of level 1. The third paper exhibits an interesting link between Maeda's conjecture and the factoricaton of eigenforms of level 1.
    
    
    1. (with Ken Ono), On the irreducibility of Hecke polynomials, Journal of Number Theory, 73 (1998), 527--532.
    2. (with David Farmer), The irreducibility of some level--1 Hecke polynomials, Math. Comp. 71 (2002) no. 239, 1263--1270.
    3. (with H. Xue and PhD Student: J. Beyerl), On the divisibility of Eigenforms by other Eigenforms, Proc. Amer. Math. Soc. (to appear).

  • Other Interests: Partition Functions, Class Number Forumlas and Cryptography

    1. (with N. Calkin and REU Students: J. Davis, K. James, E. Perez, C. Swannack), Computing the integer partition function., Math. Comp. 76 (2007), 1619-1638.
    2. (with N. Calkin and Graduate Students: T. Flowers, E. Smith and REU Students: B. Brown, A. Stout), Elliptic Curves, Modular Forms, and Sums of Hurwitz Class Numbers., Journal of Number Theory, 128, no. 6, (2008), 1847--1863.
    3. (with N. Calkin, D. Penniston; Graduate Student: N. Drake and REU Students: S. Law, P. Lee, J. Radder), Divisibility properties of the 5-regular and 13-regular partition functions, INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY, 8(1) (2008) #A60.
    4. (with N. Calkin, S. Gao, H. Maharaj; Graduate Student: E. Smith and REU Students: J. Burkhart, J. C. Hyde-Volpe, S. Manber, J. Ruiz), Finite field elements of high order arising from modular curves, Designs, Codes and Cryptography, 51:3 June 2009.

  • Complete List of Publications (with links to preprints)

• External Funding

  • Research (REU and Standard)

    One NSF standard grant and 5 REU grants totalling $1,190,652 have funded faculty and grad student summer salary and travel. The REU grants also funded undergrad summer research and undergrad travel.
    • Collaborative Research: Research Experience for Undergraduates: Algebraic geometry, combinatorics, and number theory, NSF, CoPI, $240,789, ($72,237) 2012-2014.
    • REU Site: Computation, Combinatorics and Number Theory , NSF, PI, $559,816 ($279,908), 2006-2011.
    • Supplement to 2003 REU in Computational Number Theory and Combinatorics , NSF, PI, $28,389 ($14,195), 2006-2007.
    • 2003 REU in Computational Number Theory and Combinatorics , NSF, PI, $245,556 ($122,778), 2003-2006.
    • REU in Computational Number Theory and Combinatorics , NSF, PI, $64,109 ($32,055), 2002-2003.
    • Modular Forms and Related Topics , NSF, PI, $51,993 ($51,993), 2000-2003.

  • Conference Grants

    I have served as a CoPI on several conference grants funded by NSF and NSA and totalling $96,443 which have supported the PAlmetto Number Theory Series (PANTS) as well as some meetings of the SouthEast Regional Meeting On Numbers (SERMON).

  • Scientific Computing

    I served as a CoPI on two NSF SCREMS grants totalling $272,196 which provided my department with scientific computing platforms.

  • Complete Listing of Grant Funding

• Research Advising

  • Current Advisees

    • Liem Ngyuen, (MS Math. Sci.) May 2013, coadvised with Hui Xue.
    • Jason Hedetniemi, (PhD Math. Sci.), May 2016.

  • PhD Students

    1. Bryan Faulkner, ``Estimates related to the arithmetic of elliptic curves,'' (August 2007).
    2. Ethan Smith, ``On Elliptic Curves, Modular Forms, and the Distribution of Primes'' (May 2009).
    3. Jeff Beyerl, ``On factoring Hecke eigenforms, nearly holomorphic modular forms, and applications to L-values,'' (May 2012), coadvised with Hui Xue.
    4. Catherine Trentacoste, ``Modular Forms, Elliptic Curves and Drinfeld Modules,'' (May 2012), coadvised with Hui Xue.

  • Masters Students

    1. Travieso Gonzalez, ``Exploring the partition function,'' (May 2003).
    2. Ethan Smith, ``Bases of modular forms,'' (May 2005).
    3. Matthew J. Lafferty, ``Building squares of ideals in number fields'' (May 2008).
    4. Jeff Beyerl, ``Binary Quadratic Forms over $\mathbb F[T]$ and Principal Ideal Domains,'' (May 2009), coadvised with Hui Xue..
    5. Catherine Trentacoste, ``Construction of a dimension two rank one Drinfeld Module,'' (May 2009), coadvised with Hui Xue..
    6. Rodney Keaton, ``Explicit Level Lowering for 2-Dimensional Modular Galois Representations,'' (December 2010), coadvised with Jim Brown.
    7. Jason Hedetniemi, ``Champion Primes For Elliptic Curvers,'' (May 2012), coadvised with Hui Xue..

  • Undergraduate Students

    During the course of our REU program (2002-2010; 2012-2014), I have helped supervise the research of around 80 students. Please see our REU web site for more information.

• Faculty Mentoring

  • Hui Xue (tenured and promoted to Associate Professor in 2012)
  • Jim Brown (tenured and promoted to Associate Professor in 2012)