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Explicit bounds on canonical Green functions of modular curves  [PDF]
Peter Bruin
Mathematics , 2012,
Abstract: We prove explicit bounds on canonical Green functions of Riemann surfaces obtained as compactifications of quotients of the upper half-plane by Fuchsian groups.
Explicit computations of Siegel modular forms of degree two  [PDF]
Martin Raum,Nathan C. Ryan,Nils-Peter Skoruppa,Gonzalo Tornaría
Mathematics , 2012,
Abstract: Unlike classical modular forms, there is currently no general way to implement the computation of Siegel modular forms of arbitrary weight, level and character, even in degree two. There is however, a way to do it in a unified way. After providing a survey of known computations we describe the implementation of a class modeling Siegel modular forms of degree two in Sage. In particular, we describe algorithms to compute a variety of rings of Siegel modular forms, many of which are implemented in our class. A wide variety of Siegel modular forms (e.g., both vector- and scalar-valued) can be modeled via this class and we unify these via a construct we call a formal Siegel modular form. We define this notion and discuss it in detail.
Integral points of a modular curve of level 11  [PDF]
René Schoof,Nikos Tzanakis
Mathematics , 2011,
Abstract: Using lower bounds for linear forms in elliptic logarithms we determine the integral points of the modular curve associated to the normalizer of a non-split Cartan group of level 11. As an application we obtain a new solution of the class number one problem for complex quadratic fields.
Sturm bounds for Siegel modular forms of degree 2 and odd weights  [PDF]
Toshiyuki Kikuta,Sho Takemori
Mathematics , 2015,
Abstract: We correct the proof of the theorem in the previous paper presented by the first named author, which concerns Sturm bounds for Siegel modular forms of degree $2$ and of even weights modulo a prime number dividing $2\cdot 3$. We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.
Computing modular equations for Shimura curves  [PDF]
Yifan Yang
Mathematics , 2012,
Abstract: In the classical setting, the modular equation of level $N$ for the modular curve $X_0(1)$ is the polynomial relation satisfied by $j(\tau)$ and $j(N\tau)$, where $j(\tau)$ is the standard elliptic $j$-function. In this paper, we will describe a method to compute modular equations in the setting of Shimura curves. The main ingredient is the explicit method for computing Hecke operators on the spaces of modular forms on Shimura curves developed in [13].
Bounds of some parameters of elliptic curve on finite field  [PDF]
Alexey Milovanov
Mathematics , 2013,
Abstract: I prove lower bounds of some parameters of elliptic curve over finite field. There parameters are closely interrelated with cryptographic stability of elliptic curve.
Local solubility and height bounds for coverings of elliptic curves  [PDF]
Tom Fisher,Graham Sills
Mathematics , 2011,
Abstract: We study genus one curves that arise as 2-, 3- and 4-coverings of elliptic curves. We describe efficient algorithms for testing local solubility and modify the classical formulae for the covering maps so that they work in all characteristics. These ingredients are then combined to give explicit bounds relating the height of a rational point on one of the covering curves to the height of its image on the elliptic curve. We use our results to improve the existing methods for searching for rational points on elliptic curves.
Explicit Families Of Elliptic Curves With The Same Mod 6 Representation  [PDF]
Zexiang Chen
Mathematics , 2014,
Abstract: Let $E$ be an elliptic curve over $\mathbb{Q}$. In this paper we study two certain modular curves which parameterize families of elliptic curves which are directly (resp. reverse) 6-congruent to $E$ together with the explicit parametrizations. The equations for the direct case has been worked out but the parametrization is only given for very few cases. In this paper we use a new method to obtain the equations for both direct and reverse cases together with full (and simpler) parametrizations.
Evaluating modular forms on Shimura curves  [PDF]
Paul D. Nelson
Mathematics , 2012,
Abstract: Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quotient of the hyperbolic plane, so that classical q-expansions are not available. The method takes the form of an explicit, rapidly-convergent formula that is well-suited for numerical computation. We apply it to the problem of computing modular parametrizations of elliptic curves, and illustrate with some numerical examples.
Rankin--Eisenstein classes and explicit reciprocity laws  [PDF]
Guido Kings,David Loeffler,Sarah Livia Zerbes
Mathematics , 2015,
Abstract: We construct three-variable $p$-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated $L$-value does not vanish.
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