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Jim Brown

Publications and source records attributed to Jim Brown.

7 recordsLinked to original sources
arXiv2025

Klingen Eisenstein series congruences and modularity

We construct a mod $\ell$ congruence between a Klingen Eisenstein series (associated to a classical newform $\phi$ of weight $k$) and a Siegel cusp form $f$ with irreducible Galois representation. We use this congruence to show non-vanishing of the Bloch-Kato Selmer group $H^1_f(\mathbf{Q}, \textrm{ad}^0\rho_{\phi}(2-k)\otimes \mathbf{Q}_{\ell}/\mathbf{Z}_{\ell})$ under certain assumptions and provide an example. We then prove an $R=dvr$ theorem for the Fontaine-Laffaille universal deformation ring of $\overline{\rho}_f$ under some assumptions, in particular, that the residual Selmer group $H^1_f(\mathbf{Q}, \textrm{ad}^0\overline{\rho}_{\phi}(k-2))$ is cyclic. For this we prove a result about extensions of Fontaine-Laffaille modules. We end by formulating conditions for when $H^1_f(\mathbf{Q}, \textrm{ad}^0\overline{\rho}_{\phi}(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.

math.NT↗
arXiv2024

Lightweight Safety Classification Using Pruned Language Models

In this paper, we introduce a novel technique for content safety and prompt injection classification for Large Language Models. Our technique, Layer Enhanced Classification (LEC), trains a Penalized Logistic Regression (PLR) classifier on the hidden state of an LLM's optimal intermediate transformer layer. By combining the computational efficiency of a streamlined PLR classifier with the sophisticated language understanding of an LLM, our approach delivers superior performance surpassing GPT-4o and special-purpose models fine-tuned for each task. We find that small general-purpose models (Qwen 2.5 sizes 0.5B, 1.5B, and 3B) and other transformer-based architectures like DeBERTa v3 are robust feature extractors allowing simple classifiers to be effectively trained on fewer than 100 high-quality examples. Importantly, the intermediate transformer layers of these models typically outperform the final layer across both classification tasks. Our results indicate that a single general-purpose LLM can be used to classify content safety, detect prompt injections, and simultaneously generate output tokens. Alternatively, these relatively small LLMs can be pruned to the optimal intermediate layer and used exclusively as robust feature extractors. Since our results are consistent on different transformer architectures, we infer that robust feature extraction is an inherent capability of most, if not all, LLMs.

cs.CL↗
arXiv2016

Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts

In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form $f$ on the unitary group $U(n,n)(A_F)$ for a large class of totally real fields $F$ via a divisibility of a special value of the standard $L$-function associated to $f$. We also study $\ell$-adic properties of the Fourier coefficients of an Ikeda lift $I_{\phi}$ (of an elliptic modular form $\phi$) on $U(n,n)(A_{\mathbf{Q}})$ proving that they are $\ell$-adic integers which do not all vanish modulo $\ell$. Finally we combine these results to show that the condition of $\ell$ being a congruence prime for $I_{\phi}$ is controlled by the $\ell$-divisibility of a product of special values of the symmetric square $L$-function of $\phi$. We close the paper by computing an example when our main theorem applies.

math.NT↗
arXiv2015

On the norms of $p$-stabilized elliptic newforms (with an appendix by Keith Conrad)

Let $f \in S_κ(Γ_0(N))$ be a Hecke eigenform at $p$ with eigenvalue $λ_f(p)$ for a prime $p$ not dividing $N$. Let $α_p$ and $β_p$ be complex numbers satisfying $α_p + β_p = λ_f(p)$ and $α_p β_p = p^{κ-1}$. We calculate the norm of $f_{p}^{α_p}(z) = f(z) - β_{p} f(pz)$ as well as the norm of $U_p f$, both classically and adelically. We use these results along with some convergence properties of the Euler product defining the symmetric square L-function of $f$ to give a `local' factorization of the Petersson norm of $f$.

math.NT↗
arXiv2007

On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.

math.NT↗
arXiv2005

Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture

Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f), then the prime divides the order of the Selmer group associated to f. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel eigenform that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in the Selmer group.

math.NT↗