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Non-conservative plane quartic curves in characteristic five. (English) Zbl 0718.14019

The author studies geometrically irreducible non-singular projective curves in characteristic 5 which become elliptic curves under some constant field extension. These curves are plane quartic curves and they are defined by explicit equations. There are described two invariants which are a full list of invariants of these curves. It is proved that at least one of then should be transcendental over \({\mathbb{F}}_ 5\).

MSC:

14H25 Arithmetic ground fields for curves
14G15 Finite ground fields in algebraic geometry
14H45 Special algebraic curves and curves of low genus
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[1] ANGERMÜLLER, G.,Die Wertehalbgruppe einer ebenen irreduziblen algebroiden Kurve, Math. Z.153 (1977), 267–282 · Zbl 0338.14006 · doi:10.1007/BF01214480
[2] BEDOYA, H., STÖHR, K.O.,An algorithm to calculate discrete invariants of singular primes in function fields, J. of Number Theory27 (1987), 310–323 · Zbl 0628.14027 · doi:10.1016/0022-314X(87)90070-9
[3] CHEVALLEY, C.,Introduction to the theory of algebraic functions of one variable, American Math. Society (1951), New York · Zbl 0045.32301
[4] GORENSTEIN, D.,An arithmetic theory of adjoint plane curves, Trans. Amer. Math. Soc.25 (1952), 414–436 · Zbl 0046.38503 · doi:10.1090/S0002-9947-1952-0049591-8
[5] HUSEMÖLLER, D.,Elliptic Curves, Springer-Verlag (1986), New York, Berlin, Heidelberg · Zbl 1040.11043
[6] ROQUETTE, P.,Zur Theoric der Konstantenreduktion algebraischer Manigfaltigkeiten, J. Reine Angew. Math.200 (1958), 1–44 · Zbl 0149.39202 · doi:10.1515/crll.1958.200.1
[7] ROQUETTE, P.,Über den Singularitätsgrad von Teilringen in Funktionenkörpern, Math. Z.77 (1961), 228–240 · Zbl 0102.27801 · doi:10.1007/BF01180176
[8] ROSENLICHT, M.,Equivalence relations on algebraic curves, Ann. of Math. (2)56 (1952), 169–191 · Zbl 0047.14503 · doi:10.2307/1969773
[9] SERRE, J.-P.,Cohomologie Galoisienne, Springer-Verlag. Lecture Notes in Math.5 (1973), Berlin. Heidelberg, New York
[10] SERRE, J.-P.,Groupes algébriques et corps de classes, Hermann (1959), Paris
[11] STICHTENOTH, H.,Zur Konservativität algebraischer Funktionenkörper, J. Reine Angew. Math.301 (1978), 30–45 · Zbl 0375.12014 · doi:10.1515/crll.1978.301.30
[12] STÖHR, K.O.,On singular primes in function fields, Arch. Math.50 (1988), 156–163 · Zbl 0637.12008 · doi:10.1007/BF01194574
[13] STÖHR, K.O., VILLELA, M.L.T.,Non-conservative function fields of genus (p+1)/2, Manuscripta Math. to appear · Zbl 0699.14031
[14] STÖHR, K.O., VOLOCH, J.F.,Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3)52 (1986), 1–19 · Zbl 0593.14020 · doi:10.1112/plms/s3-52.1.1
[15] STÖHR, K.O., VOLOCH, J.F.,A formula for the Cartier operator on plane algebraic curves, J. Reine Angew. Math.377 (1987), 49–64 · Zbl 0605.14023 · doi:10.1515/crll.1987.377.49
[16] SZPIRO, L.,Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque86 (1981) · Zbl 0463.00009
[17] TATE, J.,Genus change in inseparable extensions of function fields, Proc. Amer. Math. Soc.3 (1952), 400–406 · Zbl 0047.03901 · doi:10.1090/S0002-9939-1952-0047631-9
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