# modular form

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## 1—10 of 18 matching pages

##### 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
• 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 $\xi(\boldsymbol{{\Gamma}})$ 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, $y\equiv x^{r}\pmod{n}$ and $1\leq y. … By the Euler–Fermat theorem (27.2.8), $x^{\phi\left(n\right)}\equiv 1\pmod{n}$; hence $x^{t\phi\left(n\right)}\equiv 1\pmod{n}$. But $y^{s}\equiv x^{rs}\equiv x^{1+t\phi\left(n\right)}\equiv x\pmod{n}$, 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 $2^{n}\not\equiv 2\pmod{n}$, then $n$ is composite. … A Carmichael number is a composite number $n$ for which $b^{n}\equiv b\pmod{n}$ for all $b\in\mathbb{N}$. …