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Register a new userModular units and cuspidal divisor classes on $X_0(n^2M)$ with $n|24$ and $M$ squarefree
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For a positive integer $N$, let $mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $mathscr C(N)(mathbb Q):=mathscr C(N)cap J_0(N)(mathbb Q)$ be its $mathbb Q$-rational subgroup. Let also $mathscr C_{mathbb Q}(N)$ be the subgroup of $mathscr C(N)(mathbb Q)$ generated by $mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $mathscr C(N)(mathbb Q)$ and $mathscr C_{mathbb Q}(N)$ are equal. To achieve this, we show that all modular units on $X_0(N)$ on such $N$ are products of functions of the form $eta(mtau+k/h)$, $mh^2|N$ and $kinmathbb Z$ and determine the necessary and sufficient conditions for products of such functions to be modular units on $X_0(N)$.
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For a positive integer $N$, let $mathscr{C}_N(mathbb{Q})$ be the rational cuspidal subgroup of $J_0(N)$ and $mathscr{C}(N)$ be the rational cuspidal divisor class group of $X_0(N)$, which are both subgroups of the rational torsion subgroup of $J_0(N)$. We prove that two groups $mathscr{C}_N(mathbb{Q})$ and $mathscr{C}(N)$ are equal when $N=p^2M$ for any prime $p$ and any squarefree integer $M$. To achieve this we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m, h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.
We show that every Fricke invariant meromorphic modular form for $Gamma_0(N)$ whose divisor on $X_0(N)$ is defined over $mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-function of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.
We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with connections to perturbation theory. Additionally, we prove the full conjecture in some particular cases using a connection to Eisenstein congruences. Finally, our methods generalise to equidistribution results for cohomology classes of finite volume quotients of $n$-dimensional hyperbolic space.
Let $f(x)inmathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $nge 1$ and $kge 2$. An integer $a$ is called an $f$-exunit in the ring $mathbb{Z}_n$ of residue classes modulo $n$ if $gcd(f(a),n)=1$. In this paper, we use the principle of cross-classification to derive an explicit formula for the number ${mathcal N}_{k,f,c}(n)$ of solutions $(x_1,...,x_k)$ of the congruence $x_1+...+x_kequiv cpmod n$ with all $x_i$ being $f$-exunits in the ring $mathbb{Z}_n$. This extends a recent result of Anand {it et al.} [On a question of $f$-exunits in $mathbb{Z}/{nmathbb{Z}}$, {it Arch. Math. (Basel)} {bf 116} (2021), 403-409]. We derive a more explicit formula for ${mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.
We show that for some $kle 3570$ and all $k$ with $442720643463713815200|k$, the equation $phi(n)=phi(n+k)$ has infinitely many solutions $n$, where $phi$ is Eulers totient function. We also show that for a positive proportion of all $k$, the equation $sigma(n)=sigma(n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao and PolyMath.
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