Modular curves of composite level
Modular equations of hyperelliptic X₀(N) and an application
Modular forms and class number congruences
Modular functions, elliptic functions and Galois module structure.
Modular parametrizations of certain elliptic curves
Kaneko and Sakai (2013) recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and Ramanujan-Serre differential operator. In this paper, we study certain properties of the modular parametrization associated to the elliptic curves over ℚ, and as a consequence we generalize and explain some of their findings.
Modularity of fibres in rigid local systems.
Modularity of the Rankin-Selberg -series, and multiplicity one for .
Modulární křivky a Fermatova věta
Modules de Swan et courbes elliptiques à multiplication complexe
Mordell-Weil groups of generic Abelian varieties.
Mordell-Weil lattices in characteristic 2. III: A Mordell-Weil lattice of rank 128.
Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples
We study the family of elliptic curves y² = x(x-a²)(x-b²) parametrized by Pythagorean triples (a,b,c). We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over ℚ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil...
More cubic surfaces violating the Hasse principle
We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.
Motives for Hilbert modular forms.
Motives for modular forms.
Motives over totally real fields and -adic -functions
Special values of certain functions of the type are studied where is a motive over a totally real field with coefficients in another field , andis an Euler product running through maximal ideals of the maximal order of andbeing a polynomial with coefficients in . Using the Newton and the Hodge polygons of one formulate a conjectural criterium for the existence of a -adic analytic continuation of the special values. This conjecture is verified in a number of cases related to...
Motivic L-functions and Galois module structures.
Multiple zeta values and periods of moduli spaces
We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces of Riemann spheres with marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle...
Multiplicative independence and bounded height
Multivariable polynomial injections on rational numbers