Narrow Results

J.-P. Serre asserted a precise form of Torelli Theorem for genus 3 curves, namely, an indecomposable principally polarized abelian threefold is a Jacobian if and only if some specific invariant is a square. We study here a three dimensional family of such threefolds, introduced by Howe, Leprevost and Poonen. By a new formulation, we link their results to Serre's assersion. Then, we recover a formula of Klein related to the question for complex threefolds. In this case the invariant is a modular form of weight 18, and the result is proved using theta functions identities.

In this article, we investigate congruences satisfied by Apéry-like numbers.

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p₁, p₂, p₃) by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit τ → N ∈ ℤ. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.

In this paper, we study the Diophantine equation ${(x-d)}^4+x^4+{(x+d)}^4=y^n$, and based on Frey–Hellegouarch curves and their corresponding Galois representations, we solve the equation for various choices of $d$.

We find out for which t shells of selfdual lattices and of their shadows are spherical t-designs. The method uses theta series of lattices, which are modular forms. We analyze fully cubic and Witt lattices, as well as all selfdual lattices of rank at most 24.

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each k ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes ℓ such that the fixed field of the mod-ℓ representation attached to the unique normalized cusp eigenform of weight k on SL₂(ℤ) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes ℓ for each k in the list. Finally, given a non-CM curve E/ℚ, we show that there exists an integer M_E such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to M_E.

We show that a positive proportion of the Poincaré series do not vanish identically when either the index or the weight varies over an interval of suitable length, the other one being fixed.

A partition of a positive integer $n$ is called $\ell$-regular if none of its parts is divisible by $\ell$. Let $b_{11}(n)$ denote the number of 11-regular partitions of $n$. In this paper we give a complete description of the behavior of $b_{11}(n)$ modulo $5$ when $5\nmid n$ in terms of the arithmetic of the ring $\mathbb{Z}[\sqrt{-33}]$. This description is obtained by relating the generating function for these values of $b_{11}(n)$ to a Hecke eigenform, and as a byproduct we find exact criteria for which of these values are divisible by 5 in terms of the prime factorization of $12n+5$.

Special arithmetic series $f(x)={\displaystyle {\sum }_{n=0}^{\infty }{c}_n{x}^n}$, whose coefficients c_n are normally given as certain binomial sums, satisfy “self-replicating” functional identities. For example, the equation