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Let F be a totally real number field which is unramified at 3 , 5 , and 7. We give explicit conditions on a finite place v 0 of F such that any elliptic curve over F with potential good reduction at v 0 is modular. The main point is to prove that there exist no elliptic curves with reducible mod 5, and 7 representations (or mod 3, 5, and 7 representations) under our assumptions.

We define affinoids in the Lubin–Tate curve in even equal characteristic, and compute the reductions of them. Each reduction is isomorphic to a smooth affine curve with Artin–Schreier type equation. We expect that the cohomology of the reductions realizes the local Langlands correspondence and local Jacquet–Langlands correspondence for representations of conductor five.

Let X be a smooth projective algebraic variety over a number field F , with an embedding τ : F ↪ C . The action of Gal ( F ¯ / F ) on l -adic cohomology groups H e t i ( X / F ¯ , Q l ) , induces Galois representations ρ l i : Gal ( F ¯ / F ) → GL ( H e t i ( X / F ¯ , Q l ) ) . Fix a non-archimedean valuation v on F , of residual characteristic p . Let F v be the completion of F at v and ′ W v be the Weil–Deligne group of F v . We establish new cases, for which the linear representations ρ l i _ of ′ W v , associated to ρ l i , form a compatible system of representations of ′ W v defined over Q . Under suitable hypotheses, we show that in some cases, these representations actually form a compatible system of representations of ′ W v , with values in the Mumford–Tate group of H B i ( τ X ( C ) , Q ) . When X has good reduction at v , we establish a motivic relationship between the compatibility of the system { ρ l i } l ≠ p and the conjugacy class of the crystalline Frobenius of the reduction of X at v .

Let K be a number field and let ϕ in K ( z ) be a rational function of degree d ≥ 2 . Let S be the set of places of bad reduction for ϕ (including the archimedean places). Let Per ( ϕ , K ) , PrePer ( ϕ , K ) , and Tail ( ϕ , K ) be the set of K-rational periodic, preperiodic, and purely preperiodic points of ϕ, respectively. The present paper presents two main results. The first result is a bound for | PrePer ( ϕ , K ) | in terms of the number of places of bad reduction | S | and the degree d of the rational function ϕ. This bound significantly improves a previous bound given by J. Canci and L. Paladino. For the second result, assuming that | Per ( ϕ , K ) | ≥ 4 (resp. | Tail ( ϕ , K ) | ≥ 3 ), we prove bounds for | Tail ( ϕ , K ) | (resp. | Per ( ϕ , K ) | ) that depend only on the number of places of bad reduction | S | (and not on the degree d). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when | Per ( ϕ , K ) | 4 (resp. | Tail ( ϕ , K ) | 3 ).

Fix an elliptic curve E / Q . For each prime p of good reduction, let a p = p + 1 − # E ( F p ) . A well-known theorem of Hasse asserts that | a p | ≤ 2 p . We say that p is a champion prime for E if a p = − ⌊ 2 p ⌋ , that is, # E ( F p ) is as large as allowed by the Hasse bound. Analogously, we call p a trailing prime if a p = ⌊ 2 p ⌋ . In this note, we study the frequency of champion and trailing primes for CM elliptic curves. Our main theorem is that for CM curves, both the champion primes and trailing primes have counting functions ∼ 2 3 π x 3 / 4 / log ⁡ x , as x → ∞ . This confirms (in corrected form) a recent conjecture of James–Tran–Trinh–Wertheimer–Zantout.

We consider the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic curves of everywhere good reduction and rational j-invariant. Using this, we determine the density of such real and imaginary quadratic fields. If R ( X ) denotes the number of real quadratic fields K = Q [ m ] such that | Δ K | X and for which there exists an elliptic curve E / K with rational j-invariant that has everywhere good reduction, then R ( X ) ≫ X log ⁡ ( X ) . We also obtain a similar result for imaginary quadratic fields. To obtain these estimates we explicitly construct quadratic fields over which we can construct elliptic curves with everywhere good reduction. The estimates then follow from elementary multiplicative number theory. In addition, we obtain infinite families of real and imaginary quadratic fields such that there are no elliptic curves with everywhere good reduction over these fields.

Let E be an elliptic curve defined over Q. It is known that the structure of the reduction E(Fp) is (1)E(Fp)≃Z/dpZ⊕Z/epZ with dp|ep. The constant CE,j=∑k=1∞μ(k)[Q(E[jk]):Q] appears as the density of primes p with good reduction for E and dp=j (under the GRH in the non-CM case, unconditionally in the CM case). We give appropriate conditions for this constant to be positive when j>1.

Let R be a complete discrete valuation F q -algebra with fraction field K and perfect residue field k . For an irreducible smooth affine curve C , with field of constants F q , let M denote a τ -sheaf over C K , endowed with a characteristic morphism ι : Spec K→C . Given a Tate module T l ( M ) with trivial action of the inertia group I K , we construct a good model M for M over C R . This yields an analog for τ -sheaves of the classical Neron–Ogg–Shafarevic theorem on good reduction of abelian varieties. We can actually extend this result to a criterion for nondegenerate and semistable reduction. As an application, we show how the local L -factor of a τ -sheaf at a place of bad reduction is related to the action of Frobenius on the associated Galois representations. Finally, we discuss the implications of these results to Drinfeld modules and their associated t -motives.

The purpose of this paper is to generalize some results of Bloch [3] concerning class field theory for curves over local fields. Bloch developed his theory mainly under the assumption that curves have good reduction. By using a different method, we shall develope a theory including the bad reduction case.

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials.

LetCbe a hyperelliptic curve of genus 3, with potentially good reduction at all primes not dividing 3, defined over a number fieldK. A function defined on the JacobianJofCwill be constructed which, when evaluated at the 3-torsion pointsJ[3] ofJ, will giveS-units inK(J[3]); at most, the setSwill consist of primes dividing 3.