By using new techniques with the degenerate Whittaker functions found by Ikeda-Yamana, we construct higher level cusp form on $E_{7,3}$, called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercuspidal local components. This generalizes the previous results on level one forms. But there are new phenomena in higher levels; first, we can handle non-trivial central characters. Second, the lift depends only on the restriction of the Hecke cusp form to $SL_2$. Hence any twist of the cusp form gives rise to the same lift. However for square free levels with the trivial central character, there is no such ambiguity.
In this paper, we obtain the central limit theorem of Hecke eigenvalues in very general setting of split simple algebraic groups over $\mathbb{Q}$, using irreducible characters of compact Lie groups.
Let $k$ be a field of characteristic zero containing a primitive fifth root of unity. Let $X/k$ be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group $D_5$ is a subgroup of ${\rm Aut}(X)$. We find that the intermediate Jacobian $J(X)$ of $X$ is isogenous to the product of an elliptic curve $E$ and the self-product of an abelian surface $B$ with real multiplication by $\mathbb{Q}(\sqrt{5})$. We give explicit models of some algebraic curves related to the construction of $J(X)$ as a Prym variety. This includes a two parameter family of curves of genus 2 whose Jacobians are isogenous to the abelian surfaces mentioned as above.
"River's Edge" is the title of a series of collage artworks created from images obtained via the Internet through the medium of generative programming. In this series of images, the artist used a keyword associated with his childhood memory "River's Edge"to conduct an in-depth search for and gather associated images, which he then assembled into vivid and visually appealing collages. In the "River's Edge" artwork, blue, gray, and green sections of the collected images were associated with water, stone, and sky. Pieces from hundreds of images were extracted from the Internet in data form, processed, and emplaced in the images. The creative process was based on an algorithm that examined the collected images, extracted appealing sections, subjected them to a limited set of modifications, and then emplaced them into the artwork. Although the algorithm's functionalities are limited to magnification, rotation, and choosing the areas to extract from the collected imagery, the process made it possible to create a wide variety of collages. In the numerous trials that were conducted to develop this art form, several new expressions were identified, and many beautiful patterns were created.
Let $N$ be a positive integer, and $X_{0}(N)$ be the coarse moduli space that classifies a pair $(E, C)$ where $E$ is an elliptic curve and $C$ is a cyclic subgroup of order $N$ . The space $X_{0}(N)$ has a canonical structure of algebraic curve over $\mathbb{Q}$ . Let $J_{0}(N)$ be the Jacobian variety of $X_{0}(N)$ . The number of $\mathbb{Q}$-simple factors in $J_{0}(N)$ relates to the conjecture that the Mordell-Weil rank of elliptic curves over $\mathbb{Q}$ is unbounded when curves vary arbitrarily (see [4]). So it seems to be interesting to investigate $\mathbb{Q}$-simple factors of $J_{0}(N)$ . Our purpose in this paper is to determine all levels $N$ for which all $\mathbb{Q}$-simple factors of $J_{0}(N)$ are elliptic curves. As a result, we have the following Theorem.
We prove the existence of a potentially diagonalizable lift of a given automorphic mod $p$ Galois representation $\overline{\rho}:{\rm Gal}(\overline{F}/F)\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ for any totally real field $F$ and any rational prime $p>2$ under the adequacy condition by using automorphic lifting techniques developed by Barnet-Lamb, Gee, Geraghty, and Taylor. As an application, when $p$ is split completely in $F$, we prove a variant of Serre's weight conjecture for $\overline{\rho}$. The formulation of our Serre conjecture is done by following Toby Gee's philosophy. Applying these results to the case when $F=\mathbb{Q}$ with a detailed study of potentially diagonalizable, crystalline lifts with some prescribed properties, we also define classical (naive) Serre's weights. This weight would be the minimal weight among possible classical weights in some sense which occur in candidates of holomorphic Siegel Hecke eigen cusp forms of degree 2 with levels prime to $p$. The main task is to construct a potentially ordinary automorphic lift for $\overline{\rho}$ by assuming only the adequacy condition. The main theorems in this paper also extend many results obtained by Barnet-Lamb, Gee and Geraghty for potentially ordinary lifts and Gee and Geraghty for companion forms.
