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Journal articles on the topic 'P-adic logarithmic forms'

Author: Grafiati

Published: 7 September 2024

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1

Yu, Kunrui. "p-adic logarithmic forms and group varieties II." Acta Arithmetica 89, no. 4 (1999): 337–78. http://dx.doi.org/10.4064/aa-89-4-337-378.

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2

YU, KUNRUI. "P-adic logarithmic forms and group varieties I." Journal für die reine und angewandte Mathematik (Crelles Journal) 1998, no. 502 (September 15, 1998): 29–92. http://dx.doi.org/10.1515/crll.1998.090.

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3

GROSSEKLONNE, E. "Sheaves of bounded p-adic logarithmic differential forms." Annales Scientifiques de l’École Normale Supérieure 40, no. 3 (May 2007): 351–86. http://dx.doi.org/10.1016/j.ansens.2007.04.001.

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4

Iovita, Adrian, and Michael Spiess. "Logarithmic differential forms on p -adic symmetric spaces." Duke Mathematical Journal 110, no. 2 (November 2001): 253–78. http://dx.doi.org/10.1215/s0012-7094-01-11023-5.

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5

Yu, Kunrui. "p-adic logarithmic forms and a problem of Erdős." Acta Mathematica 211, no. 2 (2013): 315–82. http://dx.doi.org/10.1007/s11511-013-0106-x.

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6

LE, DANIEL, SHELLY MANBER, and SHRENIK SHAH. "ON p-ADIC PROPERTIES OF TWISTED TRACES OF SINGULAR MODULI." International Journal of Number Theory 06, no. 03 (May 2010): 625–53. http://dx.doi.org/10.1142/s1793042110003101.

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We prove that logarithmic derivatives of certain twisted Hilbert class polynomials are holomorphic modular forms modulo p of filtration p + 1. We derive p-adic information about twisted Hecke traces and Hilbert class polynomials. In this framework, we formulate a precise criterion for p-divisibility of class numbers of imaginary quadratic fields in terms of the existence of certain cusp forms modulo p. We explain the existence of infinite classes of congruent twisted Hecke traces with fixed discriminant in terms of the factorization of the associated Hilbert class polynomial modulo p. Finally, we provide a new proof of a theorem of Ogg classifying those p for which all supersingular j-invariants modulo p lie in Fp.

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7

Yu, Kunrui. "Linear forms in p-adic logarithms." Acta Arithmetica 53, no. 2 (1989): 107–86. http://dx.doi.org/10.4064/aa-53-2-107-186.

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8

Lauder, Alan G. B. "Computations with classical and p-adic modular forms." LMS Journal of Computation and Mathematics 14 (August 1, 2011): 214–31. http://dx.doi.org/10.1112/s1461157011000155.

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AbstractWe present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms, using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.

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9

BUGEAUD, YANN. "Linear forms in p-adic logarithms and the Diophantine equation formula here." Mathematical Proceedings of the Cambridge Philosophical Society 127, no. 3 (November 1999): 373–81. http://dx.doi.org/10.1017/s0305004199003692.

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10

HIRATA-KOHNO, Noriko, and Rina TAKADA. "LINEAR FORMS IN TWO ELLIPTIC LOGARITHMS IN THE p-ADIC CASE." Kyushu Journal of Mathematics 64, no. 2 (2010): 239–60. http://dx.doi.org/10.2206/kyushujm.64.239.

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11

BUGEAUD, YANN. "Effective irrationality measures for real and p-adic roots of rational numbers close to 1, with an application to parametric families of Thue–Mahler equations." Mathematical Proceedings of the Cambridge Philosophical Society 164, no. 1 (September 27, 2016): 99–108. http://dx.doi.org/10.1017/s0305004116000864.

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AbstractWe show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a large power of p divides a−b. As an application, we solve completely certain families of Thue–Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.

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12

Çokoksen, Tuba, and Murat Alan. "On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai's Conjecture." Journal of New Theory, no. 47 (June 30, 2024): 72–84. http://dx.doi.org/10.53570/jnt.1479551.

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This study proves that the Diophantine equation $\left(9d^2+1\right)^x+\left(16d^2-1\right)^y=(5d)^z$ has a unique positive integer solution $(x,y,z)=(1,1,2)$, for all $d>1$. The proof employs elementary number theory techniques, including linear forms in two logarithms and Zsigmondy's Primitive Divisor Theorem, specifically when $d$ is not divisible by $5$. In cases where $d$ is divisible by $5$, an alternative method utilizing linear forms in p-adic logarithms is applied.

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13

Yu, Kunrui. "P-adic logarithmic forms and group varieties III." Forum Mathematicum 19, no. 2 (January 20, 2007). http://dx.doi.org/10.1515/forum.2007.009.

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14

PHAM, DUC HIEP. "WEIERSTRASS ZETA FUNCTIONS AND p-ADIC LINEAR RELATIONS." Bulletin of the Australian Mathematical Society, March 11, 2024, 1–10. http://dx.doi.org/10.1017/s0004972724000091.

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Abstract We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre p-adique de variétés de groupe’, Invent. Math.40(2) (1977), 171–193].

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15

Chim, Kwok Chi. "Lower bounds for linear forms in two p-adic logarithms." Journal of Number Theory, August 2024. http://dx.doi.org/10.1016/j.jnt.2024.07.012.

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16

Heuer, Ben. "Line bundles on rigid spaces in the v-topology." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.72.

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Abstract For a smooth rigid space X over a perfectoid field extension K of $\mathbb {Q}_p$ , we investigate how the v-Picard group of the associated diamond $X^{\diamondsuit }$ differs from the analytic Picard group of X. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence $$\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*}$$ We deduce some analyticity criteria which have applications to p-adic modular forms. For algebraically closed K, we show that the sequence is also right-exact if X is proper or one-dimensional. In contrast, we show that, for the affine space $\mathbb {A}^n$ , the image of the Hodge–Tate logarithm consists precisely of the closed differentials. It follows that, up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we use this to construct the p-adic Simpson correspondence of rank one.

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17

DARMON, HENRI, ALAN LAUDER, and VICTOR ROTGER. "STARK POINTS AND -ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE." Forum of Mathematics, Pi 3 (2015). http://dx.doi.org/10.1017/fmp.2015.7.

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Let$E$be an elliptic curve over$\mathbb{Q}$, and let${\it\varrho}_{\flat }$and${\it\varrho}_{\sharp }$be odd two-dimensional Artin representations for which${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms$f$,$g$, and$h$of respective weights two, one, and one, giving rise to$E$,${\it\varrho}_{\flat }$, and${\it\varrho}_{\sharp }$via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain$p$-adic iterated integralsattached to the triple$(f,g,h)$, which are$p$-adic avatars of the leading term of the Hasse–Weil–Artin$L$-series$L(E,{\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp },s)$when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on$E$—referred to asStark points—which are defined over the number field cut out by${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$. This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when$g$and$h$are binary theta series attached to a common imaginary quadratic field in which$p$splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing$p$-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on${\mathcal{H}}_{p}\times {\mathcal{H}}$), and extensions of$\mathbb{Q}$with Galois group a central extension of the dihedral group$D_{2n}$or of one of the exceptional subgroups$A_{4}$,$S_{4}$, and$A_{5}$of$\mathbf{PGL}_{2}(\mathbb{C})$.

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