A left $R$-module consists of an abelian group $(M,+)$ and scalar multiplication $R\times M\to M$, written $(r,m)\mapsto rm$, where $R$ is a ring with identity element $1$. For every $r,s\in R$ and $m,n\in M$, requireThe module elements can be added to one another and multiplied by scalars; a product of two module elements is not part of the definition. These conventions agree with ^[ref]. For a noncommutative ring, the side matters: a right module instead has $(mr)s=m(rs)$. Over a commutative ring, the two conventions agree.

Over a field $k$, these are exactly the vector space axioms. A nonzero $a\in k$ has a multiplicative inverse, so $am=0$ implies $m=a^{-1}(am)=0$. Over the integers, $2$ has no multiplicative inverse: $2x=1$ has no solution in $\mathbb{Z}$. In the module $\mathbb{Z}/6\mathbb{Z}$, division can also lose uniqueness: $2\overline{1}=2\overline{4}=\overline{2}$. A ring element which is a unit still acts invertibly on every module; the difficulty lies with scalars which are not units.

For a module $M$ over a commutative ring $A$, its character module is the additive-group dual into $D=\mathbb{Q}/\mathbb{Z}$, with action $(a\varphi )(m)=\varphi (am)$. The group $D$ is an injective cogenerator of abelian groups. The natural adjunction ${\mathrm{Hom}}_A(N,M^{\ast })\cong {\mathrm{Hom}}_{\mathbb{Z}}(N{\otimes }_{A}M,D)$ sends $h$ to the balanced pairing $(n,m)\mapsto h(n)(m)$.

A topological abelian group $A$ with a continuous action of a topological group $G$ by group automorphisms. Continuity means joint continuity of $G\times A\to A$. For a discrete coefficient group this is equivalent to every element of $A$ having an open stabilizer.

For a finite-dimensional algebra, the Auslander–Reiten quiver has one vertex for each isomorphism class of finite-dimensional indecomposable modules, and its arrows record irreducible module homomorphisms. A homomorphism is irreducible when it is neither a split injection nor a split surjection and every factorization forces the first map to split injectively or the second to split surjectively.

A finite-dimensional algebra has finite representation type when there are only finitely many isomorphism classes of finite-dimensional indecomposable modules. This counts module types, not the dimensions or the number of direct sums formed from them. For a block of a group algebra, finite representation type is equivalent to a cyclic defect group of a block.

For $P$ cyclic of order $q=p^n$ in characteristic $p$, its group algebra is $k[t]/(t^q)$. The nilpotent Jordan normal form classifies its indecomposable finite-dimensional modules as $V_i$, $1\le i\le q$. A module homomorphism $V_i\to V_j$ is determined by an element annihilated by $t^i$, so its dimension is $\min (i,j)$.

Here $\mathcal{O}$ is a complete discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p$. The system is splitting for the finite group and its subgroups when their simple representations over $K$ and $k$ are absolutely irreducible. It permits ordinary and modular representation theory to be compared by reduction of lattices modulo the maximal ideal.

A p-local module here is a finite direct sum of induced representations from normalizers $N_G(Q)$ of nontrivial p-subgroups $Q$. This usage includes induction from $G$ itself when $G$ normalizes such a subgroup. Green correspondence and induction on vertex order express every finite module, or lattice over a complete valuation ring, as a difference of p-local modules after adjoining projective modules.

The modular lambda function is invariant under the projective level-two principal congruence subgroup. That group acts freely, and its quotient is the level-two modular curve. Adding its three modular cusps gives a sphere, and $\lambda$ has cusp values $0,1,\infty$, with simple orders in exponential cusp coordinates. Therefore it identifies the quotient with the sphere minus those three points. Since the complex upper half-plane is simply connected, this is a universal covering map, with the projective level-two group as its deck transformation group.

Choose $4a_j+6b_j=k-12j$ with nonnegative exponents, for every $j$ with $k-12j\ge 0$ and $k-12j\ne 2$. The Eisenstein series and the modular discriminant give modular forms with integer coefficients and distinct first powers of $q$. The valence formula for the modular group shows that these form a basis. Successively subtracting their leading terms proves that integral coefficients up to the greatest such $j$ force all coefficients to be integral. This is more precise than merely spanning by arbitrary monomials in $E_4,E_6$.

A weight-$2k$ modular form descends as an invariant tensor differential. At an elliptic point of order $e$, a local invariant coordinate is $w=u^e$, so its descended coefficient has a pole of order at most $\lfloor k(e-1)/e\rfloor$. At infinity, $d\tau =dq/(2\pi iq)$ gives a pole of order at most $k$. Thus $D=k[\infty ]+\lfloor k/2\rfloor [i]+\lfloor 2k/3\rfloor [\rho ]$ bounds the poles. Conversely these bounds give holomorphic pullbacks. The Riemann-Roch theorem on the compactified modular curve proves finite-dimensionality; since $X(1)$ is the Riemann sphere, the dimension is $\max (0,1-k+\lfloor k/2\rfloor +\lfloor 2k/3\rfloor )$.

For even nonnegative weight, the displayed dimension follows from multiplication by the modular discriminant and the constant-coefficient map. Every nonnegative even weight except two has a product $E_4^a{E}_6^b$ of Eisenstein series with constant term one, so $\dim M_k=1+\dim M_{k-12}$ for even $k\ge 4$. The initial dimensions are one in weight zero and zero in weight two; negative and odd weights have no nonzero level-one modular forms.

For a level-one modular form of positive even weight $k$ and constant coefficient $a_0$, put $I_f(s)={\int }_1^{\infty }(f(iy)-a_0)y^{s-1}dy$. This is an entire function. Splitting the Mellin transform at one and using $f(iy)=i^k{y}^{-k}f(i/y)$ proves the displayed continuation of $(2\pi {)}^{-s}\Gamma (s)L(f,s)$. It obeys ${\Lambda }_f(s)=i^k{\Lambda }_f(k-s)$ and has possible simple poles only at zero and $k$, with residues $-a_0,i^k{a}_0$. For a cusp form both disappear.

The modular modular weight specifies the power $(cz+d{)}^k$ in a modular form's transformation law. Multiplication adds modular weights. At level one, the central matrix $-I$ excludes nonzero odd-weight forms; a nebentypus character at higher level changes this condition to $\psi (-1)=(-1{)}^k$.

The binary powering multiplication count counts arithmetic operations. The time complexity measured in operations on individual bits also depends on operand lengths. Put $L=\lfloor {\log }_{2}m\rfloor +1$, the binary length of the modulus, and let $\ell$ be the positive exponent's binary length. Stored residues use at most $L$ bits, and an unreduced product uses at most $2L$ bits.

With ordinary long multiplication and division with remainder, one modular product costs $O(L^2)$ operations on bits. Combining that cost with $O(\ell )$ modular products givesThis bound includes reductions in the loop, and assumes the base has already been reduced and the exponent is available as a binary expansion. Reading the exponent takes $O(\ell )$ additional work; initial reduction of a very long base depends on its input length, and an inverse for a negative exponent adds further work. More generally, if a modular product costs $R(L)$ operations on bits, the powering loop costs $O(\ell R(L))$. Faster multiplication and reduction methods can improve the quadratic bound.^[ref]

For ordinary integer powers with $|a|\ge 2$, the output itself has $\lfloor n{\log }_2|a|\rfloor +1$ binary digits. Even writing it takes work proportional to that length. Consequently $O(\log (n+1))$ multiplications does not imply $O(\log (n+1))$ operations on bits: the operands grow during ordinary exponentiation by squaring, whereas modular exponentiation keeps them bounded by the modulus. A squaring or multiplication by a small fixed base can also cost less than a general product, so equal counts need not mean equal running times.

A modular symbol is the relative homology class of a path between two modular cusps. Oriented paths satisfy $\{\alpha ,\beta \}+\{\beta ,\gamma \}=\{\alpha ,\gamma \}$. Continued fractions subdivide rational-endpoint paths into unimodular edges, giving a finite presentation through Manin symbols. The Hecke operators act by finite sums of transformed paths, so their matrices can be computed by rational linear algebra.

An elliptic point of a modular curve is the image of a point of the complex upper half-plane with a nontrivial stabilizer subgroup in the effective modular subgroup. For the full modular group the stabilizer subgroup orders are two and three, represented by $i$ and $e^{2\pi i/3}$. A local disk coordinate $u$ upstairs descends as $u^e$ when the stabilizer subgroup order is $e$. The central matrices $\pm I$ are factored out before counting this order.

Direct differentiation givesConsequently this transformation takes solutions of the modified Korteweg-De Vries equation to solutions of the Korteweg-De Vries equation in these sign conventions. Both summands in $u$ have the same scaling weight, so the modified-equation scaling induces $(x,t,u)\mapsto (e^{s}x,e^{3s}t,e^{-2s}u)$.

For the three corner-normalized Lax pair eigenfunctions, let $s={\mu }_3(0,0,k)$ and $S=e^{4i{k}^{3}T{\hat{\sigma }}_3}{\mu }_2(0,T,k{)}^{-1}$. Their constant connection matrices give the displayed global relation. If their second columns are $(b,a{)}^T$ and $(B,A{)}^T$, its upper-right entry is $Ab-Ba=e^{8i{k}^{3}T}{\mu }_{3,12}(0,T,k)$. The right side is analytic in the lower half-plane. In the linear limit this is the backward-sign Airy half-line global relation at Fourier parameter $2k$, coupling all three boundary traces rather than allowing three independent boundary conditions.

For $k\ne 0$, differentiation gives $dW=-2e^{-ikz-\lambda \bar{z}/(ik)}(q_{z\bar{z}}-\lambda q)dz\wedge d\bar{z}$. Thus the differential form is closed precisely for the modified Helmholtz equation $\Delta q-4\lambda q=0$. Integrating around an oriented domain gives a global relation. The primitive $\mu e^{-ikz-\lambda \bar{z}/(ik)}$ also produces compatible first-order equations whose spectral jumps reconstruct $q$.