Defining equations of $X_{0}(2^{2n})$

Following [3], and in using several results, we describe an algorithm which compute with a level N given the cardinality over Fp of the Jacobian of elliptic curves and hyperelliptic curves of genus 2 which come from X0(N). We will also sketch how to get a plane affine model for these curves.

Computational Aspects of Algebraic Curves, 2005

We compute the P SL(2, N )-module structure of the Riemann-Roch space L(D), where D is an invariant non-special divisor on the modular curve X(N ), with N ≥ 7 prime. This depends on a computation of the ramification module, which we give explicitly. These results hold for characteristic p if X(N ) has good reduction mod p and p does not divide the order of P SL(2, N ). We give as examples the cases N = 7, 11, which were also computed using [GAP]. Applications to AG codes associated to this curve are considered, and specific examples are computed using [GAP] and [MAGMA].

Acta Arithmetica

We determine all modular curves X(N) (with $N\geq 7$) which are hyperelliptic or bielliptic. We make available a proof that the automorphism group of X(N) coincides with the normalizer of $\Gamma(N)$ in $\operatorname{PSL}_2(\mathbb{R})$.

Mathematics of Computation, 1991

We study a family of cyclic quartic fields arising from the covering of modular curves -Y,(16) -» ^(16).

Journal of Algebra, 2014

In this paper we compute the gonality over Q of the modular curve X1(N) for all N 40 and give upper bounds for each N 250. This allows us to determine all N for which X1(N) has infinitely points of degree 8. We conjecture that the modular units of Q(X1(N)) are freely generated by f2,. .. , f ⌊N/2⌋+1 where f k is obtained from the equation for X1(k).

Mathematics of Computation, 2002

We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π : X 1 (N) → C defined over Q and the jacobian of C is Q-isogenous to the abelian variety A f attached by Shimura to a newform f ∈ S 2 (Γ 1 (N)). We determine the corresponding newforms and present equations for all these curves.

We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.

Revista Colombiana De Matematicas, 2013

A generalized Fermat curve of type (p, n) is a closed Riemann surface S admitting a group H ∼ = Z n p of conformal automorphisms with S/H being the Riemann sphere with exactly n + 1 cone points, each one of order p. If (p − 1)(n − 1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by AutH (S) the normalizer of H in Aut(S). If p is a prime, and either (i) n = 4 or (ii) n is even and AutH (S)/H is not a non-trivial cyclic group or (iii) n is odd and AutH (S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3, 4}, then we also compute the field of moduli of S.

2009

Abstract In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic.

Journal of Mathematical Analysis and Applications, 2006

Ramanujan derived 23 beautiful eta-function identities, which are certain types of modular equations. We found more than 70 of certain types of modular equations by using Garvan's Maple q-series package. In this paper, we prove some new modular equations which we found by employing the theory of modular form and we give some applications for them.