Computing models for quotients of modular curves

We describe an algorithm for computing a $${\mathbb {Q}}$$ Q -rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining q-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves.

A note on cusp forms and representations of SL2(𝔽𝑝)

Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup \Gamma(p), 𝑝 a prime, is acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). In this note, we compute the relation between these two spaces in the weight 2 case.

Factorization Tests and Algorithms Arising from Counting Modular Forms and Automorphic Representations

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$.It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found not to be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.

Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

ON SPACES OF MODULAR FORMS OF EVEN WEIGHT

In the article we study the structure of space of cusp forms of an even weight and a level N with the help of cusp forms of minimal weight of the same level. The exact cutting is studied when each cusp form is a product of fixed function and a modular form of a smaller weight. Except the levels 17 and19 the cutting function is a multiplicative eta - product. In the common case the space f(z) M k-l(Γ0(N)) is not equal to the space Sk (Γ0(N)), the structure of additional space is competely studied. The result is depended on the value of the level modulo 12. Dimensions of spaces are calculated by the Cohen - Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

Computations with modular forms and Galois representations

This chapter discusses several aspects of the practical side of computing with modular forms and Galois representations. It starts by discussing computations with modular forms, and from there work towards the computation of polynomials that give the Galois representations associated with modular forms. Throughout, the chapter denotes the space of cusp forms of weight k, group Γ‎₁(N), and character ε‎ by Sₖ(N, ε‎).

CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

ON THE DIMENSION OF THE SPACE OF CUSP FORMS OF OCTAHEDRAL TYPE

For a prime q ≡ 3 ( mod 4) and the character [Formula: see text], we consider the subspace of the space of holomorphic cusp forms of weight one, level q and character χ that is spanned by forms that correspond to Galois representations of octahedral type. We prove that this subspace has dimension bounded by [Formula: see text] upto multiplication by a constant that depends only on ε.