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14
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 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.
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
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
Fields of moduli of hyperelliptic curves
, 2004
"... Let F be an algebraically closed field with char(F) ̸ = 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let ι be the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/〈ι 〉 is not cyclic. ..."
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Let F be an algebraically closed field with char(F) ̸ = 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let ι be the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/〈ι 〉 is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when Aut(X)/〈ι〉 is cyclic. 1
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
MODULI INTERPRETATION OF EISENSTEIN SERIES
, 2009
"... Let ℓ ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(ℓ), which make sense over any field k in which 6ℓ ̸ = 0 and that contains the ℓth roots of unity. Over the complex numbers, these forms include all holomorphic Eisenstein series on Γ(ℓ) in all weights, i ..."
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Let ℓ ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(ℓ), which make sense over any field k in which 6ℓ ̸ = 0 and that contains the ℓth roots of unity. Over the complex numbers, these forms include all holomorphic Eisenstein series on Γ(ℓ) in all weights, in a natural way. The graded ring Rℓ that is generated by our special modular forms turns out to be generated by certain forms in weight 1 that, over C, correspond to the Eisenstein series on Γ(ℓ). By a combination of algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of Lfunctions, we show that when k = C, the ring Rℓ, which is generated as a ring by the Eisenstein series of weight 1, contains all modular forms on Γ(ℓ) in weights ≥ 2. Our results give a straightforward method to produce models for the modular curve X(ℓ) defined over the ℓth cyclotomic field, using only exact arithmetic in the ℓtorsion field of a single Qrational elliptic curve E0.
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.