Narrow Results

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.

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.

In this paper, we consider modular forms f(z) whose q-series expansions ∑b(n)qⁿ have coefficients in a localized ring of algebraic integers . Extending results of Serre and Ono, we show that if f has non-positive weight, a congruence of the form b(ℓn + a) ≡ 0 (mod ν), where ν is a place over ℓ in , can hold for only finitely many primes ℓ ≥ 5. To obtain this, we establish an effective bound on ℓ in terms of the weight and the structure of f(z).

Fix a prime N, and consider the action of the Hecke operator T_N on the space of modular forms of full level and varying weight κ. The coefficients of the matrix of T_N with respect to the basis {E₄ⁱ E₆^j | 4i + 6j = κ} for can be combined for varying κ into a generating function F_N. We show that this generating function is a rational function for all N, and present a systematic method for computing F_N. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.

In this paper, we consider indivisibility of orders of Selmer groups for modular forms under quadratic twists. Then, we will give a generalization of a theorem of James–Ono and Kohnen–Ono.

The theory of elliptic curves and modular forms provides a precise relationship between the supersingular j-invariants and the congruences between modular forms. Kaneko and Zagier discuss a surprising generalization of these results in their paper on Atkin orthogonal polynomials. In this paper, we define an analog of the Atkin orthogonal polynomials for rank two Drinfel'd modules.

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

For a prime q ≡ 3 (mod 4) and the character , we consider the subspace of the space of holomorphic cusp forms of weight one, level q and character χ that is spanned by forms that correspond to Galois representations of octahedral type. We prove that this subspace has dimension bounded by upto multiplication by a constant that depends only on ε.

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.

Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x² + y²ⁿ = z³ in nonzero coprime integer unknowns x,y,z. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for n ∈ {5,31} there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for n ≤ 10⁷. Finally, we show that there are also no solutions for n ≡ -1 (mod 6).

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.

We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x³ + y³ = z³ in , conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.

We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

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 give a list of ${\mathrm{\text{PGL}}}_2({𝔽}_{\ell })$ number fields for $\ell \ge 11$ which have rational companion forms. Our list has 53 fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.

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 $12n+5$.

We formulate a notion of modular form on the double half-plane for half-integral weights and explain its relationship to the usual notion of modular form. The construction we provide is compatible with certain physical considerations due to the second author.

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient $F$, over any arithmetic progression. Namely, if $g_{a,b}(x)$ denotes the number of even coefficients of $F$ in degrees $n\equiv b$ (mod $a$) such that $n\le x$, then we show that $g_{a,b}(x)/\sqrt{x}$ is unbounded for $x$ large.

Note that our result is very close to the best bound currently known even in the special case of the partition function $p(n)$ (namely, $\sqrt{x}loglogx$, proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre’s classical theorem on the number of even values of $p(n)$, combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients.

Interestingly, even in the case of $p(n)$ first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre’s theorem.

Denote by $h=h(-p)$ the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ with $p$ prime. It is well known that $h=1$ for $p\in \{3,7,11,19,43,67,163\}$. Recently, all the solutions of the Diophantine equation $x^2+p^s=4y^n$ with $h=1$ were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation $x^2+p^s=2^r{y}^n$ in unknown integers $(x,y,s,r,n),$ where $s\ge 0$, $r\ge 3$, $n\ge 3$, $h\in \{1,2,3\}$ and $gcd(x,y)=1$. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al.

We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular $L$-values for short intervals.