scholarly journals A Divisor Formula and a Bound on the $$\mathbb {Q}$$-Gonality of the Modular Curve $$X_1(N)$$

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mark van Hoeij ◽  
Hanson Smith
Keyword(s):  
Modular Curve

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

On the modular curve X0(23)

2012 ◽  
pp. 317-345 ◽  
Author(s):  
René Schoof
Keyword(s):  
Modular Curve

Automorphisms of the Modular Curve

2005 ◽  
pp. 25-37 ◽  
Author(s):  
Peter Bending ◽  
Alan Camina ◽  
Robert Guralnick
Keyword(s):  
Modular Curve

Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

2019 ◽  
Vol 31 (3) ◽  
pp. 647-659
Author(s):  
Fu-Tsun Wei ◽  
Takao Yamazaki
Keyword(s):  
Prime Power ◽  
Analogous Result ◽  
Power Level ◽  
Modular Curves ◽  
Generalized Jacobian ◽  
Torsion Points ◽  
Modular Curve ◽  
Generalized Jacobians ◽  
Drinfeld Modular Curves ◽  
Primary Torsion

Abstract We consider the generalized Jacobian {\widetilde{J}} of the modular curve {X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the {\mathbb{Q}} -rational torsion points on {\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on {\widetilde{J}} and its Eisenstein property.

scholarly journals Functions and differentials on the non-split Cartan modular curve of level 11

2016 ◽  
Vol 86 (303) ◽  
pp. 437-454
Author(s):  
Julio Fernández ◽  
Josep González
Keyword(s):  
Modular Curve

scholarly journals NORMAL BASES FOR MODULAR FUNCTION FIELDS

2017 ◽  
Vol 95 (3) ◽  
pp. 384-392
Author(s):  
JA KYUNG KOO ◽  
DONG HWA SHIN ◽  
DONG SUNG YOON
Keyword(s):  
Galois Extension ◽  
Meromorphic Functions ◽  
Function Fields ◽  
Modular Function ◽  
Normal Basis ◽  
Modular Curve ◽  
Normal Bases ◽  
Free Element

We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let $\mathbb{C}(X(N))$ be the field of meromorphic functions on the modular curve $X(N)$ of level $N$. We construct a completely free element in the extension $\mathbb{C}(X(N))/\mathbb{C}(X(1))$ by means of Siegel functions.

Plane Quartic Twists of X(5, 3)

2007 ◽  
Vol 50 (2) ◽  
pp. 196-205
Author(s):  
Julio Fernández ◽  
Josep González ◽  
Joan-C. Lario
Keyword(s):  
Positive Integer ◽  
Galois Representation ◽  
Rational Points ◽  
Modular Curve

AbstractGiven an odd surjective Galois representation ϱ: Gℚ → PGL2(3) and a positive integer N, there exists a twisted modular curve X(N, 3)ϱ defined over ℚ whose rational points classify the quadratic ℚ-curves of degree N realizing ϱ. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N = 5.

scholarly journals The smooth locus in infinite-level Rapoport–Zink spaces

2020 ◽  
Vol 156 (9) ◽  
pp. 1846-1872
Author(s):  
Alexander B. Ivanov ◽  
Jared Weinstein
Keyword(s):  
Elliptic Curves ◽  
Open Subset ◽  
Formal Group ◽  
Dense Open Subset ◽  
Additional Structure ◽  
Connected Component ◽  
Deformation Spaces ◽  
Modular Curve ◽  
Divisible Groups ◽  
Smooth Locus

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

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