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In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane).
In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition. The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
In mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form where a, b, c, d are integers, and ad − bc = 1.
Let f(z)=q+∑n≥2a(n)qn be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, R ...
In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set {0,+/- 1,+/- ...
Proximal splitting methods are standard tools for nonsmooth optimization. While primal-dual methods have become very popular in the last decade for their flexibility, primal methods may still be preferred for two reasons: acceleration schemes are more effe ...
