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Michael Mertens

Publications and source records attributed to Michael Mertens.

6 recordsLinked to original sources
arXiv2026

Hurwitz class number relations and mock modular forms

A classical class number relation of Hurwitz expresses the Fourier coefficients of the product of a unary theta function and the class number generating function. Here, we establish an infinite family of analogous class number relations obtained by replacing the unary quadratic form $m^2$ by positive-definite binary quadratic forms. These identities involve Cohen's generalized class numbers and depend only on the genus of the underlying quadratic form. For this, we construct a genus-dependent level-lowering operator.

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arXiv2026

On $U_p$-congruences for meromorphic modular forms with supersingularity

In this paper, we investigate congruences for meromorphic modular forms $F$ which have a pole at a single point $z$ in the fundamental domain of $\mathrm{SL}_2(\mathbb Z)$. For a prime $p$ with good supersingular reduction at the elliptic curve corresponding to $z$, we show that there exists a cusp form $f$ such that $F|U_p^m \equiv f|U_p^m \pmod{p^{\kappa_m}}$, where $\kappa_m=\alpha m -\beta$ with $\alpha$ only depending on the weight of $F$ and $\beta$ depending on $F$ and $p$ but is independent of $m$. In particular, if the space of cusp forms is trivial, then $F|U_p^m\equiv 0 \pmod{p^{\kappa_m}}$ vanishes $p$-adically to a high order. In order to prove these results, we use the fact that $p$ has supersingular reduction to realize $F$ as an overconvergent modular form and then utilize the theory of overconvergent forms to show the congruences.

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arXiv2023

On Kleinian mock modular forms

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight $2$ newforms under the $\xi$-operator. Our work uses a new non-analytic completion of the Kleinian $\zeta$-function from the theory of Abelian functions.

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arXiv2023

On Jacobi--Weierstrass mock modular forms

We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq 2$ with rational coefficients under the $\xi$-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished preimages under this differential operator of weight $2$ newforms associated to rational elliptic curves using the classical Weierstrass theory of elliptic functions. We extend this theory and construct a vector-valued Jacobi--Weierstrass $\zeta$-function which is a generalization of the classical Weierstrass $\zeta$-function.

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