# Tag Info

### Properties of the ring of all holomorphic modular forms

It may be easier to consider this question for genus zero groups, where we can construct modular forms vanishing at given points in terms of Hauptmoduln. I don't know how to find the divisors in ...
Accepted

### Automorphic representation of GL(1)

Using your definition of automorphic representation, the answer to Question 1 is yes, tautologically. If $V_\omega$ is the space of automorphic forms "with character $\omega$" then by ...
Accepted

### Abscissa of convergence of the $\tau$ Dirichlet series

Consider the sum $$S(x):=\sum_{n\leq x}\frac{\tau(n)}{n^{11/2}}.$$ By Theorem 1.3 in Montgomery-Vaughan: Multiplicative number theory I, the abscissa of convergence equals \sigma_c=\frac{11}{2}+\...
### Uniqueness of the $J$ invariant
Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a ...
Firstly look at the action of $Sp(2g,\mathbb{R})$ on $\mathbb{H}_g$. As suggested in David E Speyer's answer, the action is given by $M \cdot Z = (AZ + B)(CZ + D)^{-1}$ . Then fix an element in \$\...