Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over $F_2$ discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied...

Using the geometry of the projective plane over the finite field ${𝔽}_q$, we construct a Hermitian Lorentzian lattice $L_q$ of dimension $(q^2+q+2)$ defined over a certain number ring $𝒪$ that depends on $q$. We show that infinitely many of these lattices are $p$-modular, that is, $p{L}_q^{\text{'}}=L_q$, where $p$ is some prime in $𝒪$ such that ${\left|p\right|}^2=q$.The Lorentzian lattices $L_q$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q\equiv 3mod4$ is a rational prime such that $(q^2+q+1)$ is norm of some element in $\mathbb{Q}\left[\sqrt{-q}\right]$, then we find a $2q(q+1)$ dimensional...

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.

Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k\ge 2$ and of the same level $N$, both eigenfunctions of the Hecke operators, and both normalized so that $a_1\left(f\right)=a_1\left(E\right)=1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\overline{\mathbb{Q}}$ lying over a rational prime $p\>2$), the algebraic parts of the special values $L(E,\chi ,j)$ and $L(f,\chi ,j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,$$\frac{\tau \left(\overline{\chi }\right)L(f,\chi ,j)}{{\left(2\pi i\right)}^{j-1}{\Omega }_f^{\text{sgn}\left(E\right)}}\equiv \frac{\tau \left(\overline{\chi }\right)L(E,\chi ,j)}{{\left(2\pi i\right)}^j{\Omega }_E}(mod𝔭)$$where the...

We identify the weight four newform of a modular Calabi-Yau manifold studied by Hulek and Verrill. The main obstacle is that this Calabi-Yau manifold is not rigid and has bad reduction at prime 13. Replacing the original fiber product of elliptic fibrations with a fiberwise Kummer construction we reduce the problem to studying the modularity of a rigid Calabi-Yau manifold with good reduction at primes p ≥ 5.

In this paper, we prove that the representation $\rho$ from $G_{\mathbb{Q}}$ in GL${}_2\left(ℂ\right)$ with image $A_5$ in PGL${}_2\left(A_5\right)$ corresponding to the example $16$ in [B-K] is modular. This representation has conductor $5203$ and determinant ${\chi }_{-43}$; its modularity was not yet proved, since this representation does not satisfy the hypothesis of the theorems of [B-D-SB-T] and [Tay2].

This paper is essentially the text of the author’s lecture at the 2001 Journées Arithmétiques. It addresses the problem of identifying in Galois-theoretic terms those two-dimensional, $p$-adic Galois representations associated to holomorphic Hilbert modular newforms.

In this short note we give a new approach to proving modularity of $p$-adic Galois representations using a method of $p$-adic approximations. This recovers some of the well-known results of Wiles and Taylor in many, but not all, cases. A feature of the new approach is that it works directly with the $p$-adic Galois representation whose modularity is sought to be established. The three main ingredients are a Galois cohomology technique of Ramakrishna, a level raising result due to Ribet, Diamond, Taylor,...