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Journal articles on the topic 'Frobenius trace'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Bae, Sunghan. "Statistics of traces of high powers of the Frobenius for biquadratic function fields over finite fields." International Journal of Number Theory 15, no. 08 (August 19, 2019): 1675–91. http://dx.doi.org/10.1142/s1793042119500933.

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We investigate statistics of trace of high powers of the Frobenius class for biquadratic function fields over finite fields, which generalizes the result of Lorenzo, Meleleo and Milione on the trace of the Frobenius class for biquadratic covers of the projective line. Then we compare our result with Meisner’s result on the expected value of high powers of trace of Frobenius of biquadratic curves.

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2

CAENEPEEL, S., and T. GUÉDÉNON. "FULLY BOUNDED NOETHERIAN RINGS AND FROBENIUS EXTENSIONS." Journal of Algebra and Its Applications 06, no. 02 (April 2007): 189–206. http://dx.doi.org/10.1142/s0219498807002107.

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Let i: A → R be a ring morphism, and χ: R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1R) = 1A. If R is a Frobenius A-ring, then we can define a trace map tr: A → AR. If there exists an element of trace 1 in A, then A is right FBN if and only if AR is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover results of García and del Río, and Dǎscǎlescu, Kelarev and Torrecillas on actions of group and Hopf algebras on FBN rings as special cases. We also obtain applications to extensions of Frobenius algebras, and to Frobenius corings with a grouplike element.

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3

Yun, Zhiwei, and Christelle Vincent. "Galois representations attached to moments of Kloosterman sums and conjectures of Evans." Compositio Mathematica 151, no. 1 (October 7, 2014): 68–120. http://dx.doi.org/10.1112/s0010437x14007593.

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AbstractKloosterman sums for a finite field $\mathbb{F}_{p}$ arise as Frobenius trace functions of certain local systems defined over $\mathbb{G}_{m,\mathbb{F}_{p}}$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ that comes from geometry. We also give bounds on the ramification of these Galois representations. All of this is done in the generality of Kloosterman sheaves attached to reductive groups introduced by Heinloth, Ngô and Yun [Ann. of Math. (2) 177 (2013), 241–310]. As an application, we give proofs of conjectures of Evans [Proc. Amer. Math. Soc. 138 (2010), 517–531; Israel J. Math. 175 (2010), 349–362] expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C. Vincent.

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4

Kurniadi, Edi. "KILLING FORM ALJABAR LIE FROBENIUS BERDIMENSI ≤4 UNTUK MENENTUKAN KESEMISEDERHANAANNYA." EduMatSains : Jurnal Pendidikan, Matematika dan Sains 6, no. 1 (July 1, 2021): 101–10. http://dx.doi.org/10.33541/edumatsains.v6i1.2999.

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We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

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5

Chávez, Ángel, and George Todd. "Supercharacters and mixed moments of Kloosterman sums." International Journal of Number Theory 14, no. 04 (May 2018): 1023–32. http://dx.doi.org/10.1142/s1793042118500616.

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Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

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6

Tanaka, Hiromu. "The trace map of Frobenius and extending sections for threefolds." Michigan Mathematical Journal 64, no. 2 (June 2015): 227–61. http://dx.doi.org/10.1307/mmj/1434731922.

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7

HUMPHRIES, PETER. "ON THE MERTENS CONJECTURE FOR ELLIPTIC CURVES OVER FINITE FIELDS." Bulletin of the Australian Mathematical Society 89, no. 1 (February 28, 2013): 19–32. http://dx.doi.org/10.1017/s0004972712001116.

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AbstractWe introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms of the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.

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8

SADEK, MOHAMMAD. "FORMAL GROUPS AND INVARIANT DIFFERENTIALS OF ELLIPTIC CURVES." Bulletin of the Australian Mathematical Society 92, no. 1 (May 4, 2015): 44–51. http://dx.doi.org/10.1017/s0004972715000295.

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In this paper, we find a power series expansion of the invariant differential ${\it\omega}_{E}$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we derive several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.

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9

Ishii, Noburo. "Trace of Frobenius endomorphism of an elliptic curve with complex multiplication." Bulletin of the Australian Mathematical Society 70, no. 1 (August 2004): 125–42. http://dx.doi.org/10.1017/s0004972700035875.

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Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

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10

Kadison, Lars, and Burkhard Külshammer. "Depth Two, Normality, and a Trace Ideal Condition for Frobenius Extensions." Communications in Algebra 34, no. 9 (September 2006): 3103–22. http://dx.doi.org/10.1080/00927870600650291.

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11

McCARTHY, DERMOT. "3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES." International Journal of Number Theory 06, no. 03 (May 2010): 461–70. http://dx.doi.org/10.1142/s1793042110002946.

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We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

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12

González, Luis, Antonio Suárez, and Dolores García. "Geometrical and Spectral Properties of the Orthogonal Projections of the Identity." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/435730.

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We analyze the best approximation (in the Frobenius sense) to the identity matrix in an arbitrary matrix subspace ( nonsingular, being any fixed subspace of ). Some new geometrical and spectral properties of the orthogonal projection are derived. In particular, new inequalities for the trace and for the eigenvalues of matrix are presented for the special case that is symmetric and positive definite.

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13

Perret-Gentil, Corentin. "Exponential Sums Over Finite Fields and the Large Sieve." International Mathematics Research Notices 2020, no. 20 (August 31, 2018): 7139–74. http://dx.doi.org/10.1093/imrn/rny202.

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Abstract By using a variant of the large sieve for Frobenius in compatible systems developed in [24] and [27], we obtain zero-density estimates for arguments of $\ell $-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

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14

Barman, Rupam, and Neelam Saikia. "p -Adic gamma function and the trace of Frobenius of elliptic curves." Journal of Number Theory 140 (July 2014): 181–95. http://dx.doi.org/10.1016/j.jnt.2014.01.026.

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15

McCarthy, Dermot. "The trace of Frobenius of elliptic curves and thep-adic gamma function." Pacific Journal of Mathematics 261, no. 1 (February 28, 2013): 219–36. http://dx.doi.org/10.2140/pjm.2013.261.219.

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16

Deshpande, Tanmay. "Shintani descent for algebraic groups and almost characters of unipotent groups." Compositio Mathematica 152, no. 8 (June 1, 2016): 1697–724. http://dx.doi.org/10.1112/s0010437x16007429.

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In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

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17

MEISNER, PATRICK. "Expected value of high powers of trace of frobenius of biquadratic curves over a finite field." Mathematical Proceedings of the Cambridge Philosophical Society 166, no. 3 (April 22, 2018): 543–65. http://dx.doi.org/10.1017/s0305004118000105.

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AbstractDenote ΘCas the Frobenius class of a curveCover the finite field 𝔽q. In this paper we determine the expected value of Tr(ΘCn) whereCruns over all biquadratic curves whenqis fixed andgtends to infinity. This extends work done by Rudnick [15] and Chinis [5] who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda [1] who looked at ℓ-cyclic curves, for ℓ a prime, as well as cubic non-Galois curves.

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18

Barman, Rupam, and Neelam Saikia. "Certain transformations for hypergeometric series in the p-adic setting." International Journal of Number Theory 11, no. 02 (March 2015): 645–60. http://dx.doi.org/10.1142/s1793042115500359.

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In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

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19

Goodson, Heidi. "Hypergeometric functions and relations to Dwork hypersurfaces." International Journal of Number Theory 13, no. 02 (February 7, 2017): 439–85. http://dx.doi.org/10.1142/s1793042117500269.

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We give an expression for number of points for the family of Dwork K3 surfaces over finite fields of order [Formula: see text] in terms of Greene’s finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy’s [Formula: see text]-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.

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20

Wang, Zhongwei, Yuanyuan Chen, and Liangyun Zhang. "Separable extensions for crossed products over monoidal Hom-Hopf algebras." Journal of Algebra and Its Applications 17, no. 09 (August 23, 2018): 1850161. http://dx.doi.org/10.1142/s021949881850161x.

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Let [Formula: see text] be a Frobenius monoidal Hom-Hopf algebra, and [Formula: see text] an [Formula: see text]-Hom-Hopf Galois extension of [Formula: see text]. We prove that the separability of the Hom-algebra extension [Formula: see text] is equivalent to the existence of a trace one element [Formula: see text] that centralizes [Formula: see text]. As applications, we obtain the differentiated conditions for the extension [Formula: see text] to be separable, and deduce a Doi’s result of Hom-type.

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21

JAMES, KEVIN, and ETHAN SMITH. "Average Frobenius distribution for the degree two primes of a number field." Mathematical Proceedings of the Cambridge Philosophical Society 154, no. 3 (January 16, 2013): 499–525. http://dx.doi.org/10.1017/s0305004112000631.

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AbstractLet K be a number field and r an integer. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree two prime ideals of K with trace of Frobenius equal to r. Under certain restrictions on K, we show that “on average” the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

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22

KUO, WENTANG, and YU-RU LIU. "GAUSSIAN LAWS ON DRINFELD MODULES." International Journal of Number Theory 05, no. 07 (November 2009): 1179–203. http://dx.doi.org/10.1142/s1793042109002638.

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Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

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23

He, Saiying, and J. McLaughlin. "Some remarks on the number of points on elliptic curves over finite prime field." Bulletin of the Australian Mathematical Society 75, no. 1 (February 2007): 135–49. http://dx.doi.org/10.1017/s0004972700039034.

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Let p ≥ 5 be a prime and for a, b ε p, let Ea, b denote the elliptic curve over p with equation y2 = x3 + ax + b. As usual define the trace of Frobenius ap, a, b by We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums and for primes p in various congruence classes.As an example of our results, we prove the following: Let p ≡ 5 (mod 6) be prime and let b ε *p. Then

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24

JAMES, KEVIN, and ETHAN SMITH. "Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals." Mathematical Proceedings of the Cambridge Philosophical Society 150, no. 3 (March 15, 2011): 439–58. http://dx.doi.org/10.1017/s0305004111000041.

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AbstractLet K be a fixed number field, assumed to be Galois over ℚ. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

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25

Morel, Sophie. "Cohomologie d’intersection des variétés modulaires de Siegel, suite." Compositio Mathematica 147, no. 6 (September 28, 2011): 1671–740. http://dx.doi.org/10.1112/s0010437x11005409.

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AbstractIn this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur’s invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur’s L2 trace formula and the fixed point formula of Morel [Complexes pondérés sur les compactifications de Baily–Borel. Le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61]. We ‘stabilize’ this last formula, i.e. express it as a sum of stable distributions on the general symplectic groups and its endoscopic groups, and obtain the formula conjectured by Kottwitz in [Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties and L-functions, Part I, Perspectives in Mathematics, vol. 10 (Academic Press, San Diego, CA, 1990), 161–209]. Applications of the results of this article have already been given by Kottwitz, assuming Arthur’s conjectures. Here, we give weaker unconditional applications in the cases of the groups GSp4 and GSp6.

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26

Haines, Thomas J. "On Connected Components of Shimura Varieties." Canadian Journal of Mathematics 54, no. 2 (April 1, 2002): 352–95. http://dx.doi.org/10.4153/cjm-2002-012-x.

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AbstractWe study the cohomology of connected components of Shimura varieties coming from the group GSp2g, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character ϖ on the group of connected components of we define an operator L(ω) on the cohomology groups with compact supports Hic(, ), and then we prove that the virtual trace of the composition of L(ω) with a Hecke operator f away from p and a sufficiently high power of a geometric Frobenius , can be expressed as a sum of ω-weighted (twisted) orbital integrals (where ω-weighted means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character ϖ). As the crucial step, we define and study a new invariant α1(γ0; γ, δ) which is a refinement of the invariant α(γ0; γ, δ) defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

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27

Surawy-Stepney, Trystan, Jonas Kahn, Richard Kueng, and Madalin Guta. "Projected Least-Squares Quantum Process Tomography." Quantum 6 (October 20, 2022): 844. http://dx.doi.org/10.22331/q-2022-10-20-844.

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We propose and investigate a new method of quantum process tomography (QPT) which we call projected least squares (PLS). In short, PLS consists of first computing the least-squares estimator of the Choi matrix of an unknown channel, and subsequently projecting it onto the convex set of Choi matrices. We consider four experimental setups including direct QPT with Pauli eigenvectors as input and Pauli measurements, and ancilla-assisted QPT with mutually unbiased bases (MUB) measurements. In each case, we provide a closed form solution for the least-squares estimator of the Choi matrix. We propose a novel, two-step method for projecting these estimators onto the set of matrices representing physical quantum channels, and a fast numerical implementation in the form of the hyperplane intersection projection algorithm. We provide rigorous, non-asymptotic concentration bounds, sampling complexities and confidence regions for the Frobenius and trace-norm error of the estimators. For the Frobenius error, the bounds are linear in the rank of the Choi matrix, and for low ranks, they improve the error rates of the least squares estimator by a factor d2, where d is the system dimension. We illustrate the method with numerical experiments involving channels on systems with up to 7 qubits, and find that PLS has highly competitive accuracy and computational tractability.

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28

Yin, Penghang, and Jack Xin. "PhaseLiftOff: An accurate and stable phase retrieval method based on difference of trace and Frobenius norms." Communications in Mathematical Sciences 13, no. 4 (2015): 1033–49. http://dx.doi.org/10.4310/cms.2015.v13.n4.a10.

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29

Levin, Brandon. "Local models for Weil-restricted groups." Compositio Mathematica 152, no. 12 (September 27, 2016): 2563–601. http://dx.doi.org/10.1112/s0010437x1600765x.

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We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

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30

Görtz, Ulrich. "Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models for GL4 and GL5." Journal of Algebra 278, no. 1 (August 2004): 148–72. http://dx.doi.org/10.1016/j.jalgebra.2003.07.002.

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31

Weston, Tom. "Power Residues of Fourier Coefficients of Modular Forms." Canadian Journal of Mathematics 57, no. 5 (October 1, 2005): 1102–20. http://dx.doi.org/10.4153/cjm-2005-042-5.

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AbstractLet ρ: GQ → GLn(Qℓ) be a motivic ℓ-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under ρ is an m-th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of ρ is open. We further conjecture that for such ρ the set of these primes p is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem(which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.

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32

Martin, Robert J., Ionel-Dumitrel Ghiba, and Patrizio Neff. "A polyconvex extension of the logarithmic Hencky strain energy." Analysis and Applications 17, no. 03 (May 2019): 349–61. http://dx.doi.org/10.1142/s0219530518500173.

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Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function [Formula: see text] which is equal to the classical Hencky strain energy [Formula: see text] in a neighborhood of the identity matrix 𝟙; here, [Formula: see text] denotes the set of [Formula: see text]-matrices with positive determinant, [Formula: see text] denotes the deformation gradient, [Formula: see text] is the corresponding stretch tensor, [Formula: see text] is the principal matrix logarithm of [Formula: see text], [Formula: see text] is the trace operator, [Formula: see text] is the Frobenius matrix norm and [Formula: see text] is the deviatoric part of [Formula: see text]. The extension can also be chosen to be coercive, in which case Ball’s classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions [Formula: see text] in the so-called Valanis–Landel form [Formula: see text] with [Formula: see text], where [Formula: see text] denote the singular values of [Formula: see text].

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33

Chen, Evan, Peter S. Park, and Ashvin A. Swaminathan. "Elliptic curve variants of the least quadratic nonresidue problem and Linnik’s theorem." International Journal of Number Theory 14, no. 01 (November 21, 2017): 255–88. http://dx.doi.org/10.1142/s1793042118500161.

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Let [Formula: see text] and [Formula: see text] be [Formula: see text]-nonisogenous, semistable elliptic curves over [Formula: see text], having respective conductors [Formula: see text] and [Formula: see text] and both without complex multiplication. For each prime [Formula: see text], denote by [Formula: see text] the trace of Frobenius. Assuming the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power [Formula: see text]-functions [Formula: see text] where [Formula: see text], we prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text] This improves and makes explicit a result of Bucur and Kedlaya. Now, if [Formula: see text] is a subinterval with Sato–Tate measure [Formula: see text] and if the symmetric power [Formula: see text]-functions [Formula: see text] are functorial and satisfy GRH for all [Formula: see text], we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text]

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34

Finkel, Federico, and Artemio González-López. "The open Haldane–Shastry chain: thermodynamics and criticality." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 9 (September 1, 2022): 093102. http://dx.doi.org/10.1088/1742-5468/ac8801.

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Abstract We study the thermodynamics and criticality of the su(m|n) Haldane–Shastry chain of BC N type with a general chemical potential term. We first derive a complete description of the spectrum of this model in terms of BC N -type motifs, from which we deduce a representation for the partition function as the trace of a product of site-dependent transfer matrices. In the thermodynamic limit, this formula yields a simple expression for the free energy per spin in terms of the Perron–Frobenius eigenvalue of the continuum limit of the transfer matrix. Evaluating this eigenvalue we obtain closed-form expressions for the thermodynamic functions of the chains with m, n ⩽ 2. Using the motif-based description of the spectrum derived here, we study in detail the ground state of these models and their low energy excitations. In this way we identify the critical intervals in chemical potential space and compute their corresponding Fermi velocities. By contrast with previously studied models of this type, we find in some cases two types of low energy excitations with linear energy-quasimomentum relation. Finally, we determine the central charge of all the critical phases by analyzing the low-temperature behavior of the expression for the free energy per spin.

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35

Xie, Guojun, Huanhuan Yang, Hao Deng, Zhengpu Shi, and Gang Chen. "Formal Verification of Robot Rotary Kinematics." Electronics 12, no. 2 (January 11, 2023): 369. http://dx.doi.org/10.3390/electronics12020369.

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With the widespread application of robots in aerospace, medicine, automation, and other fields, their motion safety is essential for the well-being of humans and the accomplishment of vital socially beneficial programs. Conventional robot hardware and software designs mainly rely on experiential knowledge and manual testing to ensure safety, but this fails to cover all possible testing paths and adds risks. Alternatively, formal, mathematically rigorous verifications can provide predictable and reliable guarantees of robot motion safety. To demonstrate the feasibility of this approach, we formalize the mathematical coordinate transformation of a robot’s rigid-body kinematics using the Coq Proof Assistant to verify the correctness of its theoretical design. First, based on record-type matrix formalization, we define and verify a robot’s spatial geometry by constructing formal expressions of the matrix’ Frobenius norm, trace, and inner product. Second, we divide rotary motion into revolution and rotation construct and provide their formal definitions. Next, we formally verify the rotational matrices of angle conventions (e.g., roll–pitch–yaw and Euler), and we complete the formal verification of the Rodriguez formula to formally verify the correctness of the motion theory in specific rotating kinematics problems. The formal work of this paper has a variety of essential applications and provides a generalizable kinematics analysis framework for robot control system verification. Moreover, it paves the way for automatic programming capabilities.

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Tankeev, Sergei G. "On Frobenius traces." Izvestiya: Mathematics 62, no. 1 (February 28, 1998): 157–90. http://dx.doi.org/10.1070/im1998v062n01abeh000189.

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37

Nikolaev, Igor. "On traces of Frobenius endomorphisms." Finite Fields and Their Applications 25 (January 2014): 270–79. http://dx.doi.org/10.1016/j.ffa.2013.10.003.

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38

Shimizu, Koji. "Finiteness of Frobenius Traces of a Sheaf on a Flat Arithmetic Scheme." International Mathematics Research Notices 2020, no. 9 (June 20, 2018): 2864–80. http://dx.doi.org/10.1093/imrn/rny145.

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Abstract For a lisse $\ell $-adic sheaf on a scheme flat and of finite type over $\mathbb{Z}$, we consider the field generated over $ \mathbb{Q}$ by Frobenius traces of the sheaf at closed points. Assuming conjectural properties of geometric Galois representations of number fields and the Generalized Riemann Hypothesis, we prove that the field is finite over $\mathbb{Q}$ when the sheaf is de Rham at $\ell $ pointwise. This is a number field analog of Deligne’s finiteness result about Frobenius traces in the function field case.

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39

Bombieri, Enrico, and Nicholas Katz. "A Note on lower bounds for Frobenius traces." L’Enseignement Mathématique 56, no. 3 (2010): 203–27. http://dx.doi.org/10.4171/lem/56-3-1.

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40

González, Josep. "The Frobenius traces distribution for modular Abelian surfaces." Ramanujan Journal 33, no. 2 (December 31, 2013): 247–61. http://dx.doi.org/10.1007/s11139-013-9543-6.

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41

Bae, Sunghan, and Hwanyup Jung. "Statistics for products of traces of high powers of the Frobenius class of hyperelliptic curves in even characteristic." International Journal of Number Theory 15, no. 07 (July 21, 2019): 1519–30. http://dx.doi.org/10.1142/s1793042119500878.

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We study the averages of products of traces of high powers of the Frobenius class of real hyperelliptic curves of genus [Formula: see text] over a fixed finite field [Formula: see text] in both odd and even characteristic cases.

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42

Lennon, Catherine. "Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves." Proceedings of the American Mathematical Society 139, no. 6 (November 3, 2010): 1931–38. http://dx.doi.org/10.1090/s0002-9939-2010-10609-3.

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43

BANWAIT, BARINDER, FRANCESC FITÉ, and DANIEL LOUGHRAN. "Del Pezzo surfaces over finite fields and their Frobenius traces." Mathematical Proceedings of the Cambridge Philosophical Society 167, no. 01 (April 10, 2018): 35–60. http://dx.doi.org/10.1017/s0305004118000166.

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AbstractLet S be a smooth cubic surface over a finite field $\mathbb{F}$q. It is known that #S($\mathbb{F}$q) = 1 + aq + q2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.

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Snowden, Andrew, and Jacob Tsimerman. "Constructing elliptic curves from Galois representations." Compositio Mathematica 154, no. 10 (August 29, 2018): 2045–54. http://dx.doi.org/10.1112/s0010437x18007315.

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Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

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45

Lercier, Reynald, Christophe Ritzenthaler, Florent Rovetta, and Jeroen Sijsling. "Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields." LMS Journal of Computation and Mathematics 17, A (2014): 128–47. http://dx.doi.org/10.1112/s146115701400031x.

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AbstractWe study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

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Cho, Peter J., and Henry H. Kim. "Central limit theorem for Artin L-functions." International Journal of Number Theory 13, no. 01 (November 16, 2016): 1–14. http://dx.doi.org/10.1142/s1793042117500014.

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We show that the sum of the traces of Frobenius elements of Artin [Formula: see text]-functions in a family of [Formula: see text]-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for [Formula: see text] and [Formula: see text]-fields. We also prove a central limit theorem for the [Formula: see text]-functions of modular forms on congruence subgroups [Formula: see text] as [Formula: see text].

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47

Pacheco, Amílcar. "Distribution of the traces of Frobenius on elliptic curves over function fields." Acta Arithmetica 106, no. 3 (2003): 255–63. http://dx.doi.org/10.4064/aa106-3-4.

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Rudnick, Zeév. "Traces of high powers of the Frobenius class in the hyperelliptic ensemble." Acta Arithmetica 143, no. 1 (2010): 81–99. http://dx.doi.org/10.4064/aa143-1-5.

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Lercier, Reynald, Christophe Ritzenthaler, Florent Rovetta, Jeroen Sijsling, and Benjamin Smith. "Distributions of Traces of Frobenius for Smooth Plane Curves Over Finite Fields." Experimental Mathematics 28, no. 1 (July 11, 2017): 39–48. http://dx.doi.org/10.1080/10586458.2017.1328321.

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Barman, Rupam, and Gautam Kalita. "Hypergeometric functions over $\mathbb {F}_q$ and traces of Frobenius for elliptic curves." Proceedings of the American Mathematical Society 141, no. 10 (June 21, 2013): 3403–10. http://dx.doi.org/10.1090/s0002-9939-2013-11617-5.

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