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Gonality of modular curves in characteristic p
 Math. Res. Letters
"... Abstract. Let k be an algebraically closed field of characteristic p. Let X(p e; N) be the curve parameterizing elliptic curves with full level N structure (where p ∤ N) and full level p e Igusa structure. By modular curve, we mean a quotient of any X(p e; N) by any subgroup of ((Z/p e Z) × × SL2( ..."
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Abstract. Let k be an algebraically closed field of characteristic p. Let X(p e; N) be the curve parameterizing elliptic curves with full level N structure (where p ∤ N) and full level p e Igusa structure. By modular curve, we mean a quotient of any X(p e; N) by any subgroup of ((Z/p e Z) × × SL2(Z/NZ)) /{±1}. We prove that in any sequence of distinct modular curves over k, the kgonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X0(N) for p ∤ N. We give an application to the function field analogue of a uniform boundedness statement for the image of Galois on torsion of elliptic curves. 1.
NONHYPERELLIPTIC MODULAR CURVES OF GENUS 3
"... Abstract. A curve C defined over Q is modular of level N if there exists a nonconstant morphism X1(N) − → C defined over Q for some positive integer N. We present an algorithm to compute explicitly equations for modular nonhyperelliptic curves defined over Q of genus 3. Let C be a modular curve of ..."
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Abstract. A curve C defined over Q is modular of level N if there exists a nonconstant morphism X1(N) − → C defined over Q for some positive integer N. We present an algorithm to compute explicitly equations for modular nonhyperelliptic curves defined over Q of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between J1(N) and Jac(C) factorizes through the new part of J1(N). We compute equations of 44 nonhyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of nonnew modular curves. 1.
NONHYPERELLIPTIC MODULAR JACOBIANS OF DIMENSION 3
, 2008
"... Abstract. We present a method to solve in an efficient way the problem of constructing the curves given by Torelli’s theorem in dimension 3 over the complex numbers: For an absolutely simple principally polarized abelian threefold A over C given by its period matrix Ω, compute a model of the curve o ..."
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Abstract. We present a method to solve in an efficient way the problem of constructing the curves given by Torelli’s theorem in dimension 3 over the complex numbers: For an absolutely simple principally polarized abelian threefold A over C given by its period matrix Ω, compute a model of the curve of genus three (unique up to isomorphism) whose Jacobian, equipped with its canonical polarization, is isomorphic to A as a principally polarized abelian variety. We use this method to describe the nonhyperelliptic modular Jacobians of dimension 3. We investigate all the nonhyperelliptic new modular Jacobians Jac(Cf) of dimension 3 which are isomorphic to Af,wheref∈Snew 2 (X0(N)), N ≤ 4000.
CURVES OVER EVERY GLOBAL FIELD VIOLATING THE LOCALGLOBAL PRINCIPLE
"... Abstract. There is an algorithm that takes as input a global field k and produces a curve over k violating the localglobal principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n. 1. ..."
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Abstract. There is an algorithm that takes as input a global field k and produces a curve over k violating the localglobal principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n. 1.
ON THE ARITHMETIC OF CERTAIN MODULAR CURVES
"... Abstract. In this work, we estimate the genus of the intermediate ..."
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Abstract. In this work, we estimate the genus of the intermediate
MODULAR ISOGENY COMPLEXES
"... Abstract. We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava Etheory at height 2. Con ..."
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Abstract. We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava Etheory at height 2. Contents
Modular elliptic directions with Complex Multiplication (with an application to Gross’s elliptic curves)
, 2008
"... Let A f be the abelian variety attached by Shimura to a normalized newform f ∈ S2(Γ1(N)) and assume that A f has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in S2(Γ1(N)) corresponding to the pullbacks of the regular differentials ..."
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Let A f be the abelian variety attached by Shimura to a normalized newform f ∈ S2(Γ1(N)) and assume that A f has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in S2(Γ1(N)) corresponding to the pullbacks of the regular differentials of all elliptic quotients of A f. For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in [8]. The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then, we apply the results obtained to the case N = p 2, for primes p> 3 and p ≡ 3 mod 4. For this case we prove that if f has CM, then all optimal elliptic quotients of A f are also optimal in the sense that its endomorphism ring is the maximal order of Q ( √ −p). Moreover, if f has trivial Nebentypus then all optimal quotients are Gross’s elliptic curve A(p) and its Galois conjugates. Among all modular parametrizations J0(p 2) → A(p), we describe a canonical one and discuss some of its properties. 1
FACTORING NEWPARTS OF JACOBIANS OF CERTAIN MODULAR CURVES
"... Abstract. We prove a conjecture of Yamauchi which states that the level N for which the new part of J0(N) is Qisogenous to a product of elliptic curves is bounded. We also state and partially prove a higherdimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we deriv ..."
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Abstract. We prove a conjecture of Yamauchi which states that the level N for which the new part of J0(N) is Qisogenous to a product of elliptic curves is bounded. We also state and partially prove a higherdimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we derive a formula for the trace of Hecke operators acting on spaces S new (N, k) of newforms of weight k and level N. We use this trace formula to study the equidistribution of eigenvalues of Hecke operators on these spaces. For any d ≥ 1, we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to d. 1.