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arXiv:2404.00541math.NT

Exponential sums over singular binary quartic forms and applications

Yasuhiro Ishitsuka, Takashi Taniguchi, Frank Thorne, Stanley Yao Xiao

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Abstract

We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector spaces, but also exhibits a striking new feature: the point counting function $a_p(E) = p + 1 - \#E(\mathbb{F}_p)$ associated to an associated elliptic curve makes a prominent appearance. The proof techniques are also new, involving techniques from elementary algebraic geometry and classical invariant theory. As an application to prime number theory, we demonstrate the existence of `many' 2-Selmer elements for elliptic curves with discriminants that are squarefree and have at most four prime factors.

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math.NTNumber Theory

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  1. alphaXiv: Open research discussion on top of arXiv 1 comment

    • > Use HTML rather then PDF. The PDF is the original paper, as it appears on arXiv, so using PDF is natural. In general academics prefer PDF to HTML. In part, this is just because our tooling produces PDFs, so this is easiest. But also, we tend to prefer that the formatting be…

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  2. Ask HN: Why are we hanging on to anachronistic academic paper formats 1 comment

    • I'm an academic mathematician; feel free to ask me particular questions. Two column format is not typical in mathematics; here is an example of one of my papers, with formatting is highly typical for my field: https://arxiv.org/pdf/2404.00541.pdf I'd say that I do value…

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