In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials...

mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight ⁠1/2⁠...

example of automorphic forms. Other automorphic forms associated to these congruence subgroups are the holomorphic modular forms, which can be interpreted...

{\displaystyle \operatorname {tmf} ^{0}} (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these...

domain of Γ {\displaystyle \Gamma } . In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949. The...

half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that j ( e 2 π i / 3 ) = 0 ...

the dimension of the space of holomorphic modular forms, using the Riemann–Roch theorem (see the dimensions of modular forms). Deligne (1971) used the Eichler–Shimura...

mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex...

{\displaystyle \Delta (z)} is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol,...

locally be written as f(z)/(z − z0)n with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have essential...

its modular class vanishes. Notice that this happens if and only if there exists a volume form λ {\displaystyle \lambda } such that the modular vector...

varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of...

( S ) {\displaystyle f\in \operatorname {Diff} (S)} such that: It is holomorphic (the differential is complex linear at each point, for the structures...

functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms. Although there are numerous statements that could have...

Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3n + 1)-problem and holomorphic dynamics". Experimental Mathematics. 8 (3): 241–252. doi:10.1080/10586458...

translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can be used...

curve. Modular forms are holomorphic functions in the upper half plane characterized by their transformation properties under the modular group S L 2 (...

constraints are strong enough to almost uniquely constrain the prepotential F {\displaystyle {\mathcal {F}}} (a holomorphic function which defines the theory)...

for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers...

exponentiation is holomorphic for z ≠ 0 , {\displaystyle z\neq 0,} in the sense that its graph consists of several sheets that define each a holomorphic function...

case, one considers the algebra of holomorphic functions, i.e., complex differentiable functions. Their ratios form the field of meromorphic functions...

(CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions. While a CFT might conceivably exist only on a given Riemann...

1090/S0273-0979-1981-14936-3. Gelbart, Stephen (1977). "Automorphic forms and Artin's conjecture". Modular functions of one variable, VI (Proc. Second Internat. Conf...

as follows: K is the canonical line bundle whose sections are the holomorphic 2-forms. P n = dim ⁡ H 0 ( K n ) , n ⩾ 1 {\displaystyle P_{n}=\dim H^{0}(K^{n})...

continuous functions on V {\displaystyle V} form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are...

(or equivalently, compact), then the degree of L is determined by the holomorphic Euler characteristics of X and S: deg(L) = χ(X,OX) − 2χ(S,OS). The canonical...

ISBN 978-0-691-04955-7. Gary Cornell; Joseph H. Silverman; Glenn Stevens (2013). Modular Forms and Fermat's Last Theorem. Springer Science & Business Media. ISBN 978-1-4612-1974-3...

of forms. Fujita conjecture regarding the line bundle K M ⊗ L ⊗ m {\displaystyle K_{M}\otimes L^{\otimes m}} constructed from a positive holomorphic line...

Langlands' conjectures by reworking and expanding the classical theory of modular forms and their L-functions through the introduction of representation theory...

{\displaystyle z\mapsto \int (T_{z}f)g\,d\mu _{2}} is continuous on S and holomorphic on S for all simple functions  f  and g. For some constant k < π, the...