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DRINFELD MODULES WITH MAXIMAL GALOIS ACTION ON THEIR TORSION POINTS
DRINFELD MODULES MAXIMAL GALOIS ACTION TORSION POINTS
2015/8/26
To each Drinfeld module of generic characteristic defined over a finitely generated field, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work o...
Let E be an elliptic curve over the rationals. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of Fp-points of E is prime.However, the cons...
SPLITTING FIELDS OF CHARACTERISTIC POLYNOMIALS OF RANDOM ELEMENTS IN ARITHMETIC GROUPS
SPLITTING FIELDS CHARACTERISTIC POLYNOMIALS RANDOM ELEMENTS ARITHMETIC GROUPS
2015/8/26
We discuss rather systematically the principle, implicit in earlier works, that for a “random” element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the sp...
ABELIAN VARIETIES OVER LARGE ALGEBRAIC FIELDS WITH INFINITE TORSION
ABELIAN VARIETIES OVER LARGE ALGEBRAIC FIELDS INFINITE TORSION
2015/8/26
Let A be a non-zero abelian variety defined over a number field K and let K be a fixed algebraic closure of K. For each element σ of the absolute Galois group Gal(K/K), let K(σ) be the fixed field in ...
Consider an elliptic curve E without complex multiplication defined over the rationals.The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in term...
We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld,J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law.Mor...
Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction Av of A modulo v splits up to isogeny. Assumingthe Mumford–Tate conjec...
We show that the simple group PSL2(Fp) occurs as the Galois group of an extensionof the rationals for all primes p ≥ 5. We obtain our Galois extensions by studying the Galois action on the second étal...
ON THE SURJECTIVITY OF MOD REPRESENTATIONS ASSOCIATED TO ELLIPTIC CURVES
MOD REPRESENTATIONS ASSOCIATED ELLIPTIC CURVES
2015/8/26
Let E be an elliptic curve over the rationals that does not have complex multiplication. For each prime `, the action of the absolute Galois group on the `-torsion points of E can be given in terms of...
For a non-CM elliptic curve E/Q, Lang and Trotter made very deep conjectures concerning the number of primes p ≤ x for which ap(E) is a fixed integer (and for which the Frobeniusfield at p is a fixed ...
MODULAR FORMS AND SOME CASES OF THE INVERSE GALOIS PROBLEM
MODULAR FORMS SOME CASES INVERSE GALOIS PROBLEM
2015/8/26
We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight 3 newforms will show thatthere are Galois e...
HILBERT’S IRREDUCIBILITY THEOREM AND THE LARGER SIEVE
HILBERT IRREDUCIBILITY THEOREM LARGER SIEVE
2015/8/26
We describe an explicit version of Hilbert’s irreducibility theorem using a generalization of Gallagher’s larger sieve. We give applications to the Galois theory of random polynomials, and to the imag...
The Spider Algorithm
Spider Algorithm
2015/8/26
One of the reasons complex analytic dynamics has been such a successful subject is the deep relation that has surfaced between conformal mapping, dynamics and combinatorics. The object of the spider a...
Henon Mappings in the Complex Domain II:projective and inductive limits of polynomials
Henon Mappings Complex Domain projective and inductive limits
2015/8/26
Henon Mappings in the Complex Domain II:projective and inductive limits of polynomials.
LOCAL CONNECTIVITY OF JULIA SETS AND BIFURCATION LOCI:THREE THEOREMS OF J.-C. YOCCOZ
LOCAL CONNECTIVITY JULIA SETS BIFURCATION LOCI THREE THEOREMS OF J.-C. YOCCOZ
2015/8/26
This paper consists of three parts, each devoted to a result due to J.-C. Yoccoz.The proofs are all somewhat modied, but they are also deeply in!uenced byYoccoz's work.