Remarks on codes from modular curves: MAGMA applications

模曲线上的 Coleman 积分

Coleman Integration on Modular Curves

本文提出一种无需仿射模型的算法，利用 Hecke 算子分解积分，以计算模曲线上任意两点间的单个 Coleman 积分，并通过 SageMath 与 Magma 示例验证。

Remarks on modular symbols for Maass wave forms

In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non--positive rational ends, with values in analytic functions (pseudo--measures in the sense of [MaMar2]). After explaining the basic issues and analogies in the extended Introduction, I construct modular symbols in the sec. 1 and the related Lévy--Mellin transforms in the sec. 2. The whole paper is an extended footnote to the Lewis--Zagier fundamental study [LZ2].

A remark on non-integral p-adic slopes for modular forms

We give a sufficient condition, namely "Buzzard irregularity", for there to exist a cuspidal eigenform which does not have integral p-adic slope.

Remarks on q-exponents of generalized modular functions

In this note, we prove multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a general statement about the nature of values that the prime q-exponents of a GMF can take. We shall also study the integrality of general q-exponents of a GMF and give an upper bound on the first sign change of these q-exponents.

Remarks on the Fourier coefficients of modular forms

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that $N_p(E)=#E(\F_p)=p+1-a_p(E)$ is also a prime. We consider a variant of this question. For a newform $f$, without CM, of weight $k\geq 4$, on $Γ_0(M)$ with trivial Nebentypus $χ_0$ and with integer Fourier coefficients, let $N_p(f)=χ_0(p)p^{k-1}+1-a_p(f)$ (here $a_p(f)$ is the $p^{th}$-Fourier coefficient of $f$). We show under GRH and Artin's Holomorphy Conjecture that there …

关于通过5-剖分得到的涉及Rogers-Ramanujan连分数的模方程的一个注记

A remark on modular equations involving Rogers-Ramanujan continued fraction via $5$-dissections

本文通过研究Ramanujan theta函数的5-剖分并建立与Rogers-Ramanujan连分数相关的恒等式，重新证明了Ramanujan遗失笔记本中的一个模方程，并进一步建立了涉及$R(q)$至$R(q^{16})$的新模方程。