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,...

Special values of certain $L$ functions of the type $L(M,s)$ are studied where $M$ is a motive over a totally real field $F$ with coefficients in another field $T$, and$$L(M,s)=\prod _{𝔭}{L}_{𝔭}(M,𝒩{𝔭}^{-s})$$is an Euler product $𝔭$ running through maximal ideals of the maximal order ${𝒪}_F$ of $F$ and$$\begin{array}{cc}\hfill L_{𝔭}(M,X{)}^{-1}& =(1-{\alpha }^{\left(1\right)}(𝔭\left)X\right)·(1-{\alpha }^{\left(2\right)}(𝔭\left)X\right)·...·(1-\alpha (d\left)\right(𝔭\left)X\right)\hfill \\ & =1+A_1\left(𝔭\right)X+...+A_d\left(𝔭\right)X^d\hfill \end{array}$$being a polynomial with coefficients in $T$. Using the Newton and the Hodge polygons of $M$ one formulate a conjectural criterium for the existence of a $p$-adic analytic continuation of the special values. This conjecture is verified in a number of cases related to...

In this article we study the problem of finding such finite groups that the modular forms associated with all elements of these groups by means of a certain faithful representation belong to a special class of modular forms (so-called multiplicative $\eta -$products). This problem is open.We find metacyclic groups with such property and describe the Sylow $p$-subgroups, $p\ne 2,$ for such groups. We also give a review of the results about the connection between multiplicative $\eta$-products and elements of finite orders...