Francesc Bars

The Group Structure of the Normalizer of Γ0(N) After Atkin–Lehner

We determine the group structure of the normalizer of Γ0(N) in SL 2(ℝ) modulo Γ0(N). These results correct the Atkin–Lehner statement (Atkin and Lehner, 1970, Theorem 8).

Non-singular plane curves with an element of “large” order in its automorphism group

Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite nontrivial group) is isomorphic to a subgroup of Aut(δ) and let Mg(G) be the subset of curves δ such that G≅Aut(δ), where Aut(δ) is the full automorphism group of δ. Now, for an integer d ≥ 4, let MgPl be the subset of Mg representing smooth, genus g curves that admit a non-singular plane model of degree d (in this case, g = 1 2(d − 1)(d − 2)) and consider the sets MgPl(G) := M gPl ∩ M g(G) and MgPl(G) := M g(G) ∩ MgPl. In this paper we first determine, for an arbitrary but a fixed degree d, an algorithm to list the possible values m for which MgPl(ℤ/mℤ) is non-empty, where ℤ/mℤ denotes the cyclic group of order m. In particular, we prove that m should divide one of the integers: d − 1, d, d2 − 3d + 3, (d − 1)2, d(d − 2) or d(d − 1). Secondly, consider a curve δ ∈ MgPl with g = 1 2(d − 1)(d − 2) such that Aut(δ) has an el...

Automorphism Groups of Nonsingular Plane Curves of Degree 5

Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite nontrivial group) is isomorphic to a subgroup of Aut(δ), and let be the subset of curves δ such that G ≅ Aut(δ), where Aut(δ) is the full automorphism group of δ. Now, for an integer d ≥ 4, let be the subset of Mg representing smooth, genus g, plane curves of degree d, i.e. smooth curves that admits a plane non-singular model of degree d, (in this case, g = (d − 1)(d − 2)/2), and consider the sets and . Henn in [7] and Komiya and Kuribayashi in [10], listed the groups G for which is nonempty. In this article, we determine the loci , corresponding to nonsingular degree 5 projective plane curves, which are nonempty. Also, we present the analogy of Henns results for quartic curves concerning nonsingular plane model equations associated to these loci (see Table 2 for more details). Similar arguments can be applied to deal with higher degrees.

On the tamagawa number conjecture for hecke characters

In this paper we prove the weak local Tamagawa number conjecture for the remaining non-critical cases for the motives associated to Hecke characters � : AK ! K ∗ of [1], where K is an imaginary quadratic field with cl(K) = 1, under certain restrictions which originate mainly from the Iwasawa theory of imaginary quadratic fields.

On twists of smooth plane curves

Given a smooth curve defined over a field

On Jannsen's conjecture for Hecke characters of imaginary quadratic fields

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Arithmetic Geometry over Global Function Fields

that admits a non-singular plane model over

Bielliptic Modular Curves

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ASPECTS OF IWASAWA THEORY OVER FUNCTION FIELDS

, a fixed separable closure of

ON QUADRATIC POINTS OF CLASSICAL MODULAR CURVES

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