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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 $q$, $f(\tau )$ is analytic at $q=0$, then $f(\tau )$ is called a modular form. If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form. …

3: 23.18 Modular Transformations

… ►and $\lambda \left(\tau \right)$ is a cusp form of level zero for the corresponding subgroup of SL$(2,\mathbb{Z})$. … ► $J\left(\tau \right)$ is a modular form of level zero for SL$(2,\mathbb{Z})$. …

4: Bibliography K

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Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97966-2, MathReview, ZentralBlatt

Referenced by:

§20.10(i), §20.12(i), §20.9(iii), §22.18(iv), §23.1, §23.20(ii), §23.20(v), Ch.23.

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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 $\xi (𝚪)$ is determinate: …

6: Bibliography M

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Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

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7: Bibliography C

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

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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, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. But $y^s\equiv x^{rs}\equiv x^{1+t\varphi \left(n\right)}\equiv x(modn)$, 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 $2^n\cancel{\equiv }2(modn)$, then $n$ is composite. … ►A Carmichael number is a composite number $n$ for which $b^n\equiv b(modn)$ for all $b\in \mathbb{N}$. …