By B.H. Gross, B. Mazur
Booklet by way of Gross, B.H.
Read Online or Download Arithmetic on Elliptic Curves with Complex Multiplication PDF
Similar algebraic geometry books
During this tract, Professor Moreno develops the speculation of algebraic curves over finite fields, their zeta and L-functions, and, for the 1st time, the speculation of algebraic geometric Goppa codes on algebraic curves. one of the purposes thought of are: the matter of counting the variety of strategies of equations over finite fields; Bombieri's evidence of the Reimann speculation for functionality fields, with outcomes for the estimation of exponential sums in a single variable; Goppa's thought of error-correcting codes constituted of linear platforms on algebraic curves; there's additionally a brand new evidence of the TsfasmanSHVladutSHZink theorem.
During this quantity the writer offers a unified presentation of a few of the elemental instruments and ideas in quantity conception, commutative algebra, and algebraic geometry, and for the 1st time in a ebook at this point, brings out the deep analogies among them. The geometric perspective is under pressure during the ebook.
Birational stress is a impressive and mysterious phenomenon in higher-dimensional algebraic geometry. It seems that definite common households of algebraic forms (for instance, 3-dimensional quartics) belong to an identical class sort because the projective house yet have noticeably varied birational geometric homes.
Additional resources for Arithmetic on Elliptic Curves with Complex Multiplication
It is easy to select polynomials G and H is such a way that the ﬁbration π : QS → S has a section (and the quartic V is smooth) (examples are given in [IM]). For those special smooth quartics the construction described above gives 24 ALEKSANDR PUKHLIKOV unirationality. Note that the unirationality problem is still open for a generic threedimensional quartic. Therefore, an arbitrary smooth cubic V3 ⊂ P4 and some special smooth quartics uroth problem, however, V4 ⊂ P4 are unirational. The hardest point in solving the L¨ is proving non-rationality of these varieties, which is what was done in the papers of Clemens and Griﬃths [CG] and Iskovskikh and Manin [IM].
In order for the results, obtained in this way, to be meaningful, it is necessary for each set of regularity conditions to show that a variety of general position satisﬁes these conditions; such veriﬁcation is often non-trivial and requires special work. We will explain the methods of checking the regularity conditions. The main result, a complete proof of which is given in Chapter 3, is birational superrigidity (in particular, the absence of non-trivial structures of a rationally connected ﬁbre space, the coincidence of the groups of birational and biregular selfmaps, and, as a very particular corollary, non-rationality) of generic Fano complete intersections Vd1 ·····dk = F1 ∩ · · · ∩ Fk ⊂ PN , deg Fi = di , d1 + · · · + dk = N , of arbitrary dimension (under the restriction k < 12 dim V for the number of the deﬁning equations), and certain other natural classes of Fano varieties.
They show that the assumption of smoothness is essential for the equivalence of rational connectedness and chain rational connectedness. Counting the dimension of the set of rational curves f : P1 → PM of degree d ≥ 1 meeting at dm + 1 points a smooth hypersurface Vm ⊂ PM of degree m, 1 ≤ m ≤ M , it is easy to show that the Fano hypersurface V is rationally connected and, more generally, Fano complete intersections Vm1 ·····mk = Fm1 ∩Fm2 ∩· · ·∩Fmk ⊂ PM , where Fmi is a hypersurface of degree mi , m1 + · · · + mK ≤ M , are rationally connected.
Arithmetic on Elliptic Curves with Complex Multiplication by B.H. Gross, B. Mazur