An elementary computation of the F-pure threshold of an elliptic curve

Pagi, Gilad

doi:10.1016/j.jalgebra.2018.08.028

Cited by 3 publications

(5 citation statements)

References 27 publications

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“…(see [Pag18a] for more details.) When the indeterminant λ is understood from the context we omit it and write H{n}.…”

Section: Roots Of Deuring Polynomials In Prime Characteristicmentioning

confidence: 99%

“…It is intriguing that the value of the F -pure threshold depends on whether the cross-ratio satisfies some (Möbius transformation of) Legendre polynomial. The technique we use in the proof relies on the properties of the Deuring Polynomials as presented in [Pag18a]. While some of these properties can be deduced from known facts about Legendre polynomials, we include straightforward algebraic proofs (or cite some from [Pag18a]) in order to be self-contained.…”

Section: Introductionmentioning

confidence: 99%

“…The technique we use in the proof relies on the properties of the Deuring Polynomials as presented in [Pag18a]. While some of these properties can be deduced from known facts about Legendre polynomials, we include straightforward algebraic proofs (or cite some from [Pag18a]) in order to be self-contained.…”

Section: Introductionmentioning

confidence: 99%

“…This may be very difficult, and is related to deep theorems in number theory. For example, the case where b = c = 1 is already known as it is equivalent to the fact that there are infinitely many p's such that an elliptic curve is ordinary (see [Pag18a]). Further evidence that the conjecture is connected to ordinarity is explored in [MS11] In addition, the F -pure threshold computation in Theorem 1.1 provides an immediate alternative proof for the known corollary regarding properties of the roots of Legendre polynomials mod p: Corollary 1.5.…”

Section: Introductionmentioning

confidence: 99%

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Legendre Polynomials Roots and the $F$-Pure Threshold of bivariate Forms

Pagi

2017

Preprint

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We provide a direct computation of the F -pure threshold of degree four homogeneous polynomial in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross ratio of the roots satisfies a specific Möbius transformation of a Legendre polynomial. We then make a connection between a long lasting open question, involving the relationship between the F -pure and the log canonical threshold, and roots of Legendre polynomials over Fp.

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“…(see [Pag18a] for more details.) When the indeterminant λ is understood from the context we omit it and write H{n}.…”

Section: Roots Of Deuring Polynomials In Prime Characteristicmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Legendre Polynomials Roots and the $F$-Pure Threshold of bivariate Forms

Pagi

2017

Preprint

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“…This number is usually very difficult to compute. However, there are formulas for diagonal [Her15], binomial [Her14], Calabi-Yau [BS15], Elliptic Curves [BS15,Pag18], and quasi-homogeneous one dimensional hypersurfaces [HNnBWZ16].…”

Section: Introductionmentioning

confidence: 99%

𝐹-thresholds and test ideals of Thom-Sebastiani type polynomials

González-Villa

Jaramillo

Núñez-Betancourt

2022

Proc. Amer. Math. Soc.

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We provide a formula for F -thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. This result extends the formula for the F -pure threshold of a diagonal hypersurface. We also compute the first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our result to find hypersurfaces where the log canonical thresholds equals the F -pure thresholds for infinitely many prime numbers.Theorem A (see Theorem 3.4). Let K be a perfect field of prime characteristic p. Let R 1 = K[x 1 , . . . , x n ] and R 2 = K[y 1 , . . . , y m ] with maximal homogeneous ideals m 1 = (x 1 , . . . , x n ) and

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Legendre polynomials roots and the F-pure threshold of bivariate forms

Pagi¹

2022

J. Commut. Algebra

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An elementary computation of the F-pure threshold of an elliptic curve

Abstract: We compute the F -pure threshold of a degree three homogeneous polynomial in three variables with an isolated singularity. The computation uses elementary methods to prove a known result of Bhatt and Singh (from [2]).

Cited by 3 publications

References 27 publications

Legendre Polynomials Roots and the $F$-Pure Threshold of bivariate Forms

Legendre Polynomials Roots and the $F$-Pure Threshold of bivariate Forms

𝐹-thresholds and test ideals of Thom-Sebastiani type polynomials

Legendre polynomials roots and the F-pure threshold of bivariate forms

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