Sturm-type bounds for modular forms over functions fields

Bounds for Siegel Modular Forms of genus 2 modulo $p$

Sturm obtained the bounds for the number of the first Fourier coefficients of elliptic modular form $f$ to determine vanishing of $f$ modulo a prime $p$. In this paper, we study analogues of Sturm's bound for Siegel modular forms of genus 2. We show the resulting bound is sharp. As an application, we study congruences involving Atkin's $U(p)$-operator for the Fourier coefficients of Siegel mdoular forms of genus 2.

Congruences for Hermitan modular forms of degree 2

We give two congruence properties of Hermitian modular forms of degree 2 over $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. The one is a congruence criterion for Hermitian modular forms which is generalization of Sturm's theorem. Another is the well-definedness of the $p$-adic weight for Hermitian modular forms.

A ring of symmetric Hermitian modular forms of degree $2$ with integral Fourier coefficients

We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of generators consisting of $24$ modular forms. As an application of our structure theorem, we give the Sturm bounds of such the modular forms of weight $k$ with $4\mid k$, in the case $p=2$, $3$. We remark that the bounds for $p\ge 5$ are already known.

A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let $b \ge 2$ be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of $b$-ary digits is a Sturmian sequence over $\{0,1,\ldots, b-1\}$ and we prove that this lower bound is best possible…

具有仅非负系数的模形式

Modular Forms with Only Nonnegative Coefficients

研究SL₂(Z)上无负傅里叶系数的模形式，确定非负性Sturm界A(k)的上下界及傅里叶系数上界。

Hecke operators in half-integral weight

In \cite{Shimura}, Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm's bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^\ell}$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators $T_{p^{\ell}}$ commute with the Shimura correspondence.

Explicit Small Image Theorems for Residual Modular Representations

Let $ρ$ f,$λ$ be the residual Galois representation attached to a newform f and a prime ideal $λ$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals $λ$ such that $ρ$ f,$λ$ is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to A4, S4 or A5. We also develop explicit criteria to check the reducibility of $ρ$ f,$λ$ , leading to an algorithm that compute the exact set of such $λ$. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Ka…

Modularity of the Consani-Scholten quintic

We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over QQ, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livne method to induced four-dimensional Galois representations over QQ. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by Jose Burgos Gil and the second author.