Displaying similar documents to “Comments on the Links Between s u ( 3 ) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards”

Rational torsion points on Jacobians of modular curves

Hwajong Yoo (2016)

Acta Arithmetica

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Let p be a prime greater than 3. Consider the modular curve X₀(3p) over ℚ and its Jacobian variety J₀(3p) over ℚ. Let (3p) and (3p) be the group of rational torsion points on J₀(3p) and the cuspidal group of J₀(3p), respectively. We prove that the 3-primary subgroups of (3p) and (3p) coincide unless p ≡ 1 (mod 9) and 3 ( p - 1 ) / 3 1 ( m o d p ) .

Welschinger invariants of small non-toric Del Pezzo surfaces

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin (2013)

Journal of the European Mathematical Society

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We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at q real and s 1 pairs of conjugate imaginary points, where q + 2 s 5 , and the real quadric blown up at s 1 pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil’s recursive formula [22] for Gromov–Witten invariants of these surfaces and generalizes our recursive formula [12] for purely real Welschinger invariants...

Theta height and Faltings height

Fabien Pazuki (2012)

Bulletin de la Société Mathématique de France

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Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized Abelian variety. We also give as an application an explicit upper bound on the number of K -rational points of a curve of genus g 2 under a conjecture of S. Lang and J. Silverman. We complete the study with a comparison between differential lattice structures.

Local Indecomposability of Hilbert Modular Galois Representations

Bin Zhao (2014)

Annales de l’institut Fourier

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We prove the indecomposability of the Galois representation restricted to the p -decomposition group attached to a non CM nearly p -ordinary weight two Hilbert modular form over a totally real field F under the assumption that either the degree of F over is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of F .

On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas

Igor E. Shparlinski (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola a , p ( X , Y ) = ( x , y ) : x y a ( m o d p ) , 1 x X , 1 y Y . We give asymptotic formulas for the average values ( x , y ) a , p ( X , Y ) x y * φ ( | x - y | ) / | x - y | and ( x , y ) a , p ( X , X ) x y * φ ( | x - y | ) with the Euler function φ(k) on the differences between the components of points of a , p ( X , Y ) .

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let K be a field, a,b ∈ K and ab ≠ 0. Consider the polynomials g₁(x) = xⁿ+ax+b, g₂(x) = xⁿ+ax²+bx, where n is a fixed positive integer. We show that for each k≥ 2 the hypersurface given by the equation S k i : u ² = j = 1 k g i ( x j ) , i=1,2, contains a rational curve. Using the above and van de Woestijne’s recent results we show how to construct a rational point different from the point at infinity on the curves C i : y ² = g i ( x ) , (i=1,2) defined over a finite field, in polynomial time.

Generalised Weber functions

Andreas Enge, François Morain (2014)

Acta Arithmetica

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A generalised Weber function is given by N ( z ) = η ( z / N ) / η ( z ) , where η(z) is the Dedekind function and N is any integer; the original function corresponds to N=2. We classify the cases where some power N e evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating N ( z ) and j(z). Our ultimate goal is the use of these invariants in...

Quadratic modular symbols on Shimura curves

Pilar Bayer, Iván Blanco-Chacón (2013)

Journal de Théorie des Nombres de Bordeaux

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We introduce the concept of modular symbol and study how these symbols are related to p -adic L -functions. These objects were introduced in [] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic p -adic L -functions to more general Shimura curves.

The Brauer–Manin obstruction for curves having split Jacobians

Samir Siksek (2004)

Journal de Théorie des Nombres de Bordeaux

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Let X 𝒜 be a non-constant morphism from a curve X to an abelian variety 𝒜 , all defined over a number field k . Suppose that X is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on X to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that 𝒜 ( k ) and Ш ( 𝒜 / k ) are finite.

Gauss–Manin connections for p -adic families of nearly overconvergent modular forms

Robert Harron, Liang Xiao (2014)

Annales de l’institut Fourier

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We interpolate the Gauss–Manin connection in p -adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p -adic weight shifted by 2 . Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank...

Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

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This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve  C over  . The focus is on practical aspects of this problem in the case that the genus of  C is at least  2 , and therefore the set of rational points is finite.