Rational torsion points on Jacobians of modular curves

Hwajong Yoo

Displaying similar documents to “Rational torsion points on Jacobians of modular curves”

Visible Points on Modular Exponential Curves

Tsz Ho Chan, Igor E. Shparlinski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We obtain an asymptotic formula for the number of visible points (x,y), that is, with gcd(x,y) = 1, which lie in the box [1,U] × [1,V] and also belong to the exponential modular curves $y \equiv a g^{x} (m o d p)$ . Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

Comments on the Links Between $s u (3)$ Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards

M. Bauer, A. Coste, C. Itzykson, P. Ruelle (1997)

Recherche Coopérative sur Programme n°25

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A condition for the rationality of certain elliptic modular forms over primes dividing the level

Andrea Mori (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $f$ be a weight $k$ holomorphic automorphic form with respect to $Γ_{0} (N)$ . We prove a sufficient condition for the integrality of $f$ over primes dividing $N$ . This condition is expressed in terms of the values at particular $C M$ curves of the forms obtained by iterated application of the weight $k$ Maaß operator to $f$ and extends previous results of the Author.

The cuspidal torsion packet on hyperelliptic Fermat quotients

David Grant, Delphy Shaulis (2004)

Journal de Théorie des Nombres de Bordeaux

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Let $ℓ \geq 7$ be a prime, $C$ be the non-singular projective curve defined over $ℚ$ by the affine model $y (1 - y) = x^{ℓ}$ , $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$ , and $φ : C \to J$ the albanese embedding with $\infty$ as base point. Let $\bar{ℚ}$ be an algebraic closure of $ℚ$ . Taking care of a case not covered in [], we show that $φ (C) \cap J_{tors} (\bar{ℚ})$ consists only of the image under $φ$ of the Weierstrass points of $C$ and the points $(x, y) = (0, 0)$ and $(0, 1)$ , where $J_{tors}$ denotes the torsion points of $J$ .

Overconvergent modular forms

Vincent Pilloni (2013)

Annales de l’institut Fourier

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We give a geometric definition of overconvergent modular forms of any $p$ -adic weight. As an application, we reprove Coleman’s theory of $p$ -adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.

The second moment of quadratic twists of modular L-functions

K. Soundararajan, Matthew P. Young (2010)

Journal of the European Mathematical Society

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We study the second moment of the central values of quadratic twists of a modular $L$ -function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.

Dimensions of spaces of modular forms for $Γ_{H} (N)$

Jordi Quer (2010)

Acta Arithmetica

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Rational points on $X_{0}^{+} (p)$ .

Galbraith, Steven D. (1999)

Experimental Mathematics

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Bounds on sup-norms of half-integral weight modular forms

Eren Mehmet Kıral (2014)

Acta Arithmetica

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Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup-norm of a half-integral weight cusp form is bounded in terms of the level: we prove that $| | y^{κ / 2} {f ̃ | |}_{\infty} ≪_{ε, κ} N^{1 / 2 - 1 / 18 + ε} | | y^{κ / 2} f ̃ {| |}_{L^{2}}$ for a modular form f̃ of level 4N and weight κ, a half-integer.

Rational points on the modular curves $X_{split} (p)$

Fumiyuki Momose (1984)

Compositio Mathematica

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