arXiv subjects

Marius Fischer

Publications and source records attributed to Marius Fischer.

3 recordsLinked to original sources

arXiv2026

Diophantine rank stability and non-vanishing of $L$-functions

Let $A/\mathbb{Q}$ be a modular abelian variety of analytic rank $0$. If $G$ is a non-trivial finite abelian group such that all prime factors of $\lvert G \rvert$ are sufficiently large in terms of $A$, we show that there are infinitely many $G$-extensions $F/\mathbb{Q}$ such that $A(F)$ is finite. When $A$ is a rational elliptic curve of analytic rank zero with no exceptional primes, or the product of two such curves, the same conclusion holds without any assumptions on $|G|$. Our proof relies on new simultaneous non-vanishing results for twisted central $L$-values of even-weight holomorphic newforms. These results are obtained via novel constructions related to horizontal $p$-adic $L$-functions and are of independent interest.

math.NT↗

arXiv2026

$L^4$-norms of automorphic forms in the depth aspect

Let $p$ be an odd prime, and suppose $f$ is an $L^2$-normalised newform for $\Gamma_0(p^n)$ with bounded spectral parameters and trivial central character. We prove the optimal $L^4$-norm bound $\lVert f \rVert_4 \ll_{p,\varepsilon}(p^{n})^\varepsilon $ for all $\varepsilon >0$ as $n \rightarrow \infty$.

math.NT↗

arXiv2025

The spin of prime ideals and level-raising of even Galois representations

By extending the notion of spin of prime ideals, we show that a short character sum conjecture implies that the set of primes raising the level of a certain even Galois representation has density 2/3, as conjectured by Ramakrishna in 1998.

math.NT↗