# Questions tagged [modular-forms]

Questions about modular forms and related areas

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### Finding a rational point of large height on an elliptic curve knowing a real approximation

Let $y^2=x(x^2+n)$ be an elliptic curve with $n\in\Bbb Z$ (the same question can of course be asked for a general e.c). I know (e.g. it has rank 1) that there exists a nontrivial rational point $(r,s)$...
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### Automorphic representation of GL(1)

These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something. I am reading automorphic forms from this book. What I have understood till now: ...
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### Reference book on the relation between modular forms and elliptic curves

What is a modern reference book to understand the relation between modular forms and elliptic curves after the proof of the Taniyama–Shimura theorem?
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### Abscissa of convergence of the $\tau$ Dirichlet series

Define the $\tau$ Dirichlet series $L$ by $$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$ where $\tau$ is defined by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$ where $|q|\lt 1$....
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### When do Fourier coefficients vanish in Hida families?

Suppose you have a Hida family with $q$-expansion $F = \sum_{n=1}^{\infty} a_n(T) q^n$, where the coefficients $a_n(T)$ are power series in $\mathbb{Z}_p [[T]]$. Assume that $F$ is a cuspidal ...
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### Identity related to Ramanujan's congruences

A very simple question: how do you prove the following identity: $$\sum_{k=0}^\infty p_{5k+4}x^k=5\frac{\phi(x^5)^5}{\phi(x)^6},$$ where $$\phi(x)=\prod_{n=1}^\infty 1-x^n,$$ and $p_n$ is the ...
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