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• W4367680263 abstract "Abstract In the 1960s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato–Tate for non-CM elliptic curves). In analogy with Birch’s result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>λ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>p</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {A_{lambda}(p)} of a certain family of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {K3} surfaces <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>X</m:mi> <m:mi>λ</m:mi> </m:msub> </m:math> {X_{lambda}} with generic Picard rank 19 is the <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>O</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mn>3</m:mn> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {O(3)} distribution. This distribution, which we denote by <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>π</m:mi> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {frac{1}{4pi}f(t)} , is quite different from the semicircular distribution. It is supported on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>[</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo stretchy=false>]</m:mo> </m:mrow> </m:math> {[-3,3]} and has vertical asymptotes at <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {t=pm 1} . Here we make this result explicit. We prove that if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>5</m:mn> </m:mrow> </m:math> {pgeq 5} is prime and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo>≤</m:mo> <m:mi>a</m:mi> <m:mo><</m:mo> <m:mi>b</m:mi> <m:mo>≤</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {-3leq a<bleq 3} , then <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:mo fence=true maxsize=210% minsize=210%>|</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mi mathvariant=normal>#</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>𝔽</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>λ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>p</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=false>[</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo stretchy=false>]</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> <m:mi>p</m:mi> </m:mfrac> <m:mo>-</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>π</m:mi> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msubsup> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mi>a</m:mi> <m:mi>b</m:mi> </m:msubsup> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo rspace=4.2pt stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo fence=true maxsize=210% minsize=210%>|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mfrac> <m:mn>98.28</m:mn> <m:msup> <m:mi>p</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>4</m:mn> </m:mrow> </m:msup> </m:mfrac> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> biggl{lvert}frac{#{lambdainmathbb{F}_{p}:A_{lambda}(p)in[a,b]}}{p}-% frac{1}{4pi}int_{a}^{b}f(t),dtbiggr{rvert}leqfrac{98.28}{p^{1/4}}. As a consequence, we are able to determine when a finite field <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>𝔽</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {mathbb{F}_{p}} is large enough for the discrete histograms to reach any given height near <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {t=pm 1} . To obtain these results, we make use of the theory of Rankin–Cohen brackets in the theory of harmonic Maass forms." @default.
• W4367680263 created "2023-05-03" @default.
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• W4367680263 date "2023-05-03" @default.
• W4367680263 modified "2023-10-16" @default.
• W4367680263 title "Explicit Sato–Tate type distribution for a family of K ⁢ 3 K3 surfaces" @default.
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• W4367680263 doi "https://doi.org/10.1515/forum-2022-0272" @default.
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