Francesc Bars
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View article: The modular automorphisms of quotient modular curves
The modular automorphisms of quotient modular curves Open
We obtain the modular automorphism group of any quotient modular curve of level , with . In particular, we obtain some unexpected automorphisms of order 3 that appear for the quotient modular curves when the Atkin–Lehner involution belongs…
View article: The modular automorphisms of quotient modular curves
The modular automorphisms of quotient modular curves Open
We obtain the modular automorphism group of any quotient modular curve of level $N$, with $4,9\nmid N$. In particular, we obtain some non-expected automorphisms of order 3 that appear for the quotient modular curves when the Atkin-Lehner i…
View article: On the automorphism group of quotient modular curves
On the automorphism group of quotient modular curves Open
Altres ajuts: acords transformatius de la UAB
View article: Diophantine stability for curves over finite fields
Diophantine stability for curves over finite fields Open
We carry out a survey on curves defined over finite fields that are Diophantine stable; that is, with the property that the set of points of the curve is not altered under a proper field extension. First, we derive some general results of …
View article: Infinitely many cubic points on $X_0(N)/\langle w_d\rangle$ with $N$ square-free
Infinitely many cubic points on $X_0(N)/\langle w_d\rangle$ with $N$ square-free Open
We determine all modular curves $X_0(N)/\langle w_d\rangle$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$, when $N$ is square-free.
View article: On fake ES-irreducible components of certain strata of smooth plane sextics
On fake ES-irreducible components of certain strata of smooth plane sextics Open
In this paper, we construct the first examples of what we call fake ES-irreducible components; Definition 2.8. In our way to do so, we classify the automorphism groups of smooth plane sextics that only have automorphisms of order [Formula:…
View article: The stratification by automorphism groups of smooth plane sextic curves
The stratification by automorphism groups of smooth plane sextic curves Open
We obtain the list of automorphism groups for smooth plane sextic curves over an algebraically closed field K of characteristic p=0 or p>21. Moreover, we assign to each group a geometrically complete family over K describing its correspond…
View article: Bielliptic quotient modular curves of $X_0(N)$
Bielliptic quotient modular curves of $X_0(N)$ Open
Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ has genus at least two. We determine all pairs $(N,W_N)$ s…
View article: On fake ES-irreducibile components of certain strata of smooth plane sextics
On fake ES-irreducibile components of certain strata of smooth plane sextics Open
We construct the first examples of what we call fake ES-irreducible components; Definition 2.8. In our way to do so, we classify the automorphism groups of smooth plane sextics that only have automorphisms of order 3 or less; Theorems 2.1,…
View article: Infinitely many cubic points for $X_0^+(N)$ over $\mathbb{Q}$
Infinitely many cubic points for $X_0^+(N)$ over $\mathbb{Q}$ Open
We determine all modular curves $X_0^+(N)$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$.
View article: Fitting ideals of class groups in Carlitz–Hayes cyclotomic extensions
Fitting ideals of class groups in Carlitz–Hayes cyclotomic extensions Open
We generalize some results of Greither and Popescu to a geometric Galois cover X→Y which appears naturally for example in extensions generated by pn-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz–Ha…
View article: Infinitely many cubic points for $X_0^+(N)$ over $\mathbb Q$
Infinitely many cubic points for $X_0^+(N)$ over $\mathbb Q$ Open
We determine all modular curves X+0 (N) that admit infinitely many cubic points over the rational field Q.
View article: The automorphism group of the modular curve X* 0 (N) with square-free level
The automorphism group of the modular curve X* 0 (N) with square-free level Open
We determine the automorphism group of the modular curve X* 0 (N), obtained as the quotient of the modular curve X0(N) by the group of its Atkin-Lehner involutions, for all square-free values of N.
View article: The automorphism group of the modular curve $X_0^*(N)$ with square-free level
The automorphism group of the modular curve $X_0^*(N)$ with square-free level Open
We determine the automorphism group of the modular curve $X_0^*(N)$, obtained as the quotient of the modular curve $X_0(N)$ by the group of its Atkin-Lehner involutions, for all square-free values of $N$.
View article: Bielliptic smooth plane curves and quadratic points
Bielliptic smooth plane curves and quadratic points Open
Let [Formula: see text] be a smooth plane curve of degree [Formula: see text] defined over a global field [Formula: see text] of characteristic [Formula: see text] or [Formula: see text] (up to an extra condition on [Formula: see text]). U…
View article: Hyperelliptic parametrizations of $$\pmb {\mathbb {Q}}$$-curves
Hyperelliptic parametrizations of $$\pmb {\mathbb {Q}}$$-curves Open
For a square-free integer N, we present a procedure to compute Q-curves parametrized by rational points of the modular curve X∗0(N) when this is hyperelliptic
View article: Aspects of Iwasawa theory over function fields
Aspects of Iwasawa theory over function fields Open
We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as wel…
View article: Iwasawa main conjecture for the Carlitz cyclotomic extension and applications
Iwasawa main conjecture for the Carlitz cyclotomic extension and applications Open
We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field Fq(θ) (p is a prime of Fq[θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link bet…
View article: Hypersurface model-fields of definition for smooth hypersurfaces and their twists
Hypersurface model-fields of definition for smooth hypersurfaces and their twists Open
Given a smooth projective variety of dimension $n-1\geq 1$ defined over a perfect field $k$ that admits a non-singular hypersurface modelin $\mathbb{P}^n_{\overline{k}}$ over $\overline{k}$, a fixed algebraic closure of $k$, it does not ne…
View article: Hyperelliptic parametrizations of $\mathbb{Q}$-curves
Hyperelliptic parametrizations of $\mathbb{Q}$-curves Open
For a square-free integer $N$, we present a procedure to compute $\mathbb{Q}$-curves parametrized by rational points of the modular curve $X_0^*(N)$ when this is hyperelliptic.
View article: Bielliptic modular curves $X_0^*(N)$ with square-free levels
Bielliptic modular curves $X_0^*(N)$ with square-free levels Open
Let N=1 be a square-free integer such that the modular curve X*0(N) has genus =2. We prove that X*0(N) is bielliptic exactly for 19 values of N, and we determine the automorphism group of these bielliptic curves. In particular, we obtain t…
View article: Bielliptic modular curves $X_0^*(N)$ with square-free levels
Bielliptic modular curves $X_0^*(N)$ with square-free levels Open
Let $N\geq 1$ be a square-free integer such that the modular curve $X_0^*(N)$ has genus $\geq 2$. We prove that $X_0^*(N)$ is bielliptic exactly for $19$ values of $N$, and we determine the automorphism group of these bielliptic curves. In…
View article: The Picard group of Brauer-Severi varieties
The Picard group of Brauer-Severi varieties Open
In this paper, we provide explicit generators for the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular,we provide such generators for all Brauer-Severi surfaces. To produce these generators we use …
View article: Hypersurface model-fields of definition for smooth hypersurfaces and\n their twists
Hypersurface model-fields of definition for smooth hypersurfaces and\n their twists Open
Given a smooth projective variety of dimension $n-1\\geq 1$ defined over a\nperfect field $k$ that admits a non-singular hypersurface modelin\n$\\mathbb{P}^n_{\\overline{k}}$ over $\\overline{k}$, a fixed algebraic closure of\n$k$, it does…
View article: On quadratic points of classical modular curves
On quadratic points of classical modular curves Open
Classical modular curves are of deep interest in arithmetic geometry. In this survey we show how the work of Fumiyuki Momose is involved in order to list the classical modular curves which satisfy that the set of quadratic points over $\ma…