Let $N \geq 5$ be an integer so that the $\Gamma_1(N)$-moduli problem is representable over $\mathbb{Z}[1/N]$ (both in terms of elliptic curves/generalized elliptic curves). I am interested in Katz modular forms of this level and their $q$-expansions. From my modest understanding, there are two...

I'm trying to see that the Hecke algebra defined as $\mathbb{Q}[\text{GL}_2(\mathbb{Z}_p)\backslash \text{GL}_2(\mathbb{Q}_p)/\text{GL}_2(\mathbb{Z}_p)]$ maps to the ring of correspondences $\text{Corr}_\sim^0(M_n,M_n)$ where $M_n$ is the modular curve of elliptic curves with full $n$-torsion...

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(31i)}{\eta(i)}$ that is missing. Can someone help me solve in radical form the following equation, whose solution is the value of Dedekind's modular...

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(43i)}{\eta(i)}$ that is missing. Can someone help me solve /in radical form) the following equation, whose solution is the value of Dedekind's modular...

Let $f\in S_{k+1/2}^+(4q)$ be a newform in Kohnen’s space for $q$ an odd, square-free integer. For simplicity, assume that $k$ is even. Let $$f(z) = \sum_{\substack{n\geq 1\\ n\equiv 0,1\mod 4}}a_f(n)e(nz)$$ denote the Fourier expansion of $f$ at the cusp $\infty$. Let $D>0$ be a fundamental...

I am confused about the construction of the Hecke operators. I am defining them on $\Gamma_{1}(N)$ as $$(T_{m}f)(z) = \sum_{\substack{a,d \ge 1 \\ ad = m}}\langle a\rangle\left[\Gamma_{1}(N)\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}\Gamma_{1}(N)\right]_{k}f(z),$$ where $\langle a \rangle$ is...

Let $$ P(\tau)=\sum_{p\ \mathrm{prime}} q^p,\qquad q=e^{2\pi i\tau}. $$ The coefficients of $P(\tau)^2$ count ordered Goldbach representations: $$ P(\tau)^2=\sum_{n\ge0} r_G(n)q^n, $$ where $r_G(n)$ is the number of ordered pairs of primes $(p_1,p_2)$ such that $$ p_1+p_2=n. $$ This is formally...

Elliptic curves have L-functions that correspond to modular forms. Elliptic curves are degree 3 algebraic curves. I want to know if quadratic curves have L-functions. If they do, are these L-functions related to modular forms?

Context While working with Ramanujan's $P(q)$ function: $$P(q)=1-24\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}, \hspace{.5cm} 0<|q|<1.$$ I have found the following evaluation: $$S=1-24\sum_{n=1}^{\infty}\frac{ne^{-2\pi n/5}i^{8n/5}}{1-e^{-2\pi n/5}i^{8n/5}}=\frac{15}{\pi},\tag{1}$$ Being...