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1: Frank Garvan
His research is in the areas of q -series and modular forms, and he enjoys using MAPLE in his research. …
2: 23.15 Definitions
If, as a function of q , f ( τ ) is analytic at q = 0 , then f ( τ ) is called a modular form. If, in addition, f ( τ ) 0 as q 0 , then f ( τ ) is called a cusp form. …
3: 23.18 Modular Transformations
and λ ( τ ) is a cusp form of level zero for the corresponding subgroup of SL ( 2 , ) . … J ( τ ) is a modular form of level zero for SL ( 2 , ) . …
4: Bibliography K
  • N. Koblitz (1993) Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.
  • 5: 21.5 Modular Transformations
    The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ξ ( Γ ) is determinate: …
    6: Bibliography M
  • H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
  • 7: Bibliography C
  • G. Cornell, J. H. Silverman, and G. Stevens (Eds.) (1997) Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.
  • 8: 27.16 Cryptography
    To code a piece x , raise x to the power r and reduce x r modulo n to obtain an integer y (the coded form of x ) between 1 and n . Thus, y x r ( mod n ) and 1 y < n . … By the Euler–Fermat theorem (27.2.8), x ϕ ( n ) 1 ( mod n ) ; hence x t ϕ ( n ) 1 ( mod n ) . But y s x r s x 1 + t ϕ ( n ) x ( mod n ) , so y s is the same as x modulo n . …
    9: 23.21 Physical Applications
    §23.21 Physical Applications
    In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form ( 1 - x 2 ) ( 1 - k 2 x 2 ) . …
    §23.21(iv) Modular Functions
    Physical applications of modular functions include: …
  • String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

  • 10: 27.12 Asymptotic Formulas: Primes
    A Mersenne prime is a prime of the form 2 p - 1 . … For example, if 2 n 2 ( mod n ) , then n is composite. … A Carmichael number is a composite number n for which b n b ( mod n ) for all b . …