11,860 research outputs found
Hilbert Modular Polynomials
International audienceWe present an algorithm to compute a higher dimensional analogue of modular polynomials. This higher dimensional analogue, the 'set of Hilbert modular polynomials', concerns cyclic isogenies of principally polarised abelian varieties with maximal real multiplication by a fixed totally real number field K0. We give a proof that this algorithm is correct, and provide practical improvements and an implementation for the 2-dimensional case with K0 = Q(√ 5). We also explain applications of this algorithm to point counting, walking on isogeny graphs, and computing class polynomials
On the evaluation of modular polynomials
We present two algorithms that, given a prime ell and an elliptic curve E/Fq,
directly compute the polynomial Phi_ell(j(E),Y) in Fq[Y] whose roots are the
j-invariants of the elliptic curves that are ell-isogenous to E. We do not
assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be
adapted to handle other types of modular polynomials, and we consider
applications to point counting and the computation of endomorphism rings. We
demonstrate the practical efficiency of the algorithms by setting a new
point-counting record, modulo a prime q with more than 5,000 decimal digits,
and by evaluating a modular polynomial of level ell = 100,019.Comment: 19 pages, corrected a typo in equation (8) and added equation (9
Enumerating Colorings, Tensions and Flows in Cell Complexes
We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex , generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in for
some ) or integral (with values in ). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory
Series
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
A nowhere-zero -flow on a graph is a mapping from the edges of
to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in
any fixed orientation of , at each node the sum of the labels over the
edges pointing towards the node equals the sum over the edges pointing away
from the node. We show that the existence of an \emph{integral flow polynomial}
that counts nowhere-zero -flows on a graph, due to Kochol, is a consequence
of a general theory of inside-out polytopes. The same holds for flows on signed
graphs. We develop these theories, as well as the related counting theory of
nowhere-zero flows on a signed graph with values in an abelian group of odd
order. Our results are of two kinds: polynomiality or quasipolynomiality of the
flow counting functions, and reciprocity laws that interpret the evaluations of
the flow polynomials at negative integers in terms of the combinatorics of the
graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.
Examples of M5-Brane Elliptic Genera
We determine the modified elliptic genus of an M5-brane wrapped on various
one modulus Calabi-Yau spaces, using modular invariance together with some
known Gopakumar-Vafa invariants of small degrees. As a bonus, we find
nontrivial relations among Gopakumar-Vafa invariants of different degrees and
genera from modular invariance.Comment: 13 page
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