Related papers: A note on log-algebraicity on elliptic curves

The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here…

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.

We study the low-lying zeros of various interesting families of elliptic curve L-functions. One application is an upper bound on the average analytic rank of the family of all elliptic curves. The upper bound obtained is less than two,…

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on…

Let $k =\mathbb{F}_q$ be the finite field of $q$ elements and $E$ an elliptic curve over $k$. Let $F = k(E)$ be the function field over $E$ and let $\mathcal{O} = k[E]$ be the ring of integers. We fix the place at $\infty$ of $F$ and let…

For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL_2(Z/l^n Z). There are three exceptional prime powers l^n=2,3,4 when surjectivity of the mod l^n…

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity…

We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and of the question of twin…

We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group…

In this note, I study a comparison map between a motivic and \'{e}tale cohomology group of an elliptic curve over $\mathbb{Q}$ just outside the range of Voevodsky's isomorphism theorem. I show that the property of an appropriate version of…

We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the…

We propose three kinds of explicit formulas for the elliptic lambda function by the elliptic modular function. Further, we derive incredible cubic identities as a corollary of our explicit formulas and evaluate some singular values of the…

We introduce a type of graph integrals on elliptic curves from the heat kernel. We show that such graph integrals have modular properties under the modular group $SL(2, \Z)$, and prove the polynomial nature of the anti-holomorphic…

The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct…

We obtain eigenvalue equations satisfied by various elliptic modular graphs with five links where two of the vertices are unintegrated. Solving them leads to several non--trivial algebraic identities between these graphs.

The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$ there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is the conductor of $E$. These maps may be expressed in terms of pairs of modular…

It is known that the L-function of an elliptic curve defined over Q is given by the Mellin transform of a modular form of weight 2. Does that modular form have anything to do with string theory? In this article, we address a question along…

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier…

We classify valuations on lattice polygons with values in the ring of formal power series that commute with the action of the affine unimodular group. A typical example of such valuations is induced by the Laplace transform, but as it turns…

Linear forms in logarithms over connected commutative algebraic groups over the algebraic numbers field have been studied widely. However, the theory of linear forms in logarithms over noncommutative algebraic groups have not been developed…