Abelian Varieties Over P-Adic Fields

Abelian varieties over p-adic ﬁelds

Defence of PhD thesis Jędrzej Garnek 30.07.2020 jgarnek.faculty.wmi.amu.edu.pl/doktorat/ Motivating problem Motivating problem

Definition An abelian variety is a projective algebraic variety with an addition law, which is given by morphisms. In my thesis I considered 3 problems concerning abelian varieties: Problem #1: Local torsion problem. Problem #2: Equivariant splitting of the Hodge–de Rham exact sequence. Problem #3: Class numbers of division ﬁelds. All three problems are loosely tied to the following conjecture. Conjecture (David and Weston, 2008) Suppose that E/Q is an elliptic curve without complex multiplication. Then for almost all primes p:

E(Qp)[p] = 0. Problem #1: Local torsion of abelian varieties Local torsion problem

Problem #1 – local torsion What is the behaviour of p-torsion of abelian varieties over Qp? My results concern: local torsion of non-ordinary abelian varieties, characterisation of canonical lifts of abelian varieties in terms of local torsion, local torsion of elliptic curves with complex multiplication. Local torsion for an elliptic curve with CM

Consider the elliptic curve

E : y 2 = x3 − x

with complex multiplication by Z[i]. Theorem (J.G., 2018) Let p 5 be a prime number. The following conditions are equivalent:

1 there exists a ﬁnite extension K/Qp of degree 8 such that

E(K)[p] 6= 0,

2 2 2 p is of the form sk + sk+1, where:

s0 = 0, s1 = 1, sk+2 = 4sk+1 − sk . Canonical lifts

Let A be an ordinary abelian variety over Fp. Fix a number n ∈ Z+.

n Serre–Tate theory: the set of lifts of A to Z/p has a natural structure of a group. Definition The canonical lift modulo pn of A is the neutral element of the n group of its liftings to Z/p .

Possible applications in algorithmic algebraic number theory: counting points on elliptic curves over ﬁnite ﬁelds, computing Hilbert class polynomials, constructing hyperelliptic curves suitable for cryptography. Characterization of canonical lifts

Theorem (J.G., 2020)

Let A be an abelian variety of dimension g over Fp. Suppose that n ∼ n ⊕g A(Fp)[p ] = (Z/p )

as abelian groups. Let A be a lift of A to Zp. The scheme A is the canonical lift of A modulo pn, if and only if

n ∼ n ⊕g A(Zp)[p ] = (Z/p ) . This Theorem ties the local torsion problem to the following question. Question Let A be an abelian variety over Q. Are there inﬁnitely many primes p such that AZ/p2 is the canonical lift of AFp ? Problem #2: Equivariant splitting of the Hodge–de Rham exact sequence The de Rham cohomology

Let X be a smooth projective variety over a ﬁeld k. Definition i The ith de Rham cohomology of X , denoted HdR (X /k), is the ith sheaf hypercohomology of the de Rham complex:

2 OX → ΩX /k → ΩX /k → ...

The de Rham cohomology is related to the Hodge cohomology q p (H (X , ΩX /k )) via a spectral sequence. In particular, if X is e.g. a curve, this spectral sequence degenerates and we obtain the Hodge–de Rham exact sequence:

0 1 1 0 → H (X , ΩX /k ) → HdR (X /k) → H (X , OX ) → 0. The Hodge–de Rham exact sequence

Problem #2 – equivariant splitting... Let X be a smooth projective curve with an action of a ﬁnite group G. Does the Hodge–de Rham exact sequence

0 1 1 0 → H (X , ΩX /k ) → HdR (X /k) → H (X , OX ) → 0

of X split as a sequence of k[G]-modules?

Tait and K¨ock,2018: it does not have to split for G = Z/p.

Theorem (JG, 2019)

Suppose that there exists a point P ∈ X with non-vanishing GP,2, the second higher ramiﬁcation group. Then the Hodge–de Rham exact sequence of X does not split G-equivariantly. Example

Let X /Fp be the smooth projective curve with the aﬃne part given by the equation: y m = f (xp − x),

where f ∈ Fp[x] and p - m. ∼ Consider the group action of G = hσi = Z/p on X , given by: σ(x, y) = (x + 1, y).

Corollary (J.G., 2019) If m - deg f , the Hodge–de Rham exact sequence of X does not split G-equivariantly.

Corollary (J.G., 2019)

There doesn’t exist a lift A of Jac(X ) to W2(Fp) such that

Aut(A) =∼ Aut(Jac(X )). Problem #3: Class numbers of division fields of abelian varieties Class numbers of division fields

Definition The n-th division field of an abelian variety A over Q, denoted Q(A[n]), is the field obtained by adjoining to Q the coordinates of the n-torsion points in Q. Recall that the class number of a field measures the failure of the unique factorization in its ring of integers. Problem #3 – class numbers of division fields What is the behaviour of the class number of Q(A[n]) for a fixed abelian variety A/Q and for varying n?

for abelian varieties with complex multiplication (cf. Fukuda, Komatsu, Yamagata – 2007), for elliptic curves over Q under some additional assumptions on Galois representation and local torsion (cf. Sairaiji, Yamauchi – 2015; Hiranouchi – 2019). An estimate for class numbers

Theorem (J.G., 2018) Let r be the rank of A(Q) over End(A). If either of the following conditions holds: r > dim A,

r 1, A has good reduction at p and AFp (Fp)[p] 6= 0, then: lim # Cl( (A[pn])) = ∞. n→∞ Q Idea of the proof: n investigate the Kummer extension of Q(A[p ]), switch to a local extension to give a bound on inertia groups. Numerical Example

n Actually, I prove an estimate for # Cl(Q(A[p ])). Let me illustrate my result using a numerical example. Example Let X be the smooth projective curve over Q with the aﬃne part:

X : y 2 + (x3 + x + 1)y = x3 − x2 − 2x.

Let A = Jac(X ). Then for every prime p:

n 4n−C(p) # Cl(Q(A[p ])) p

for n = 1, 2,... and some constant C(p).

For p 6∈ {2, 3, 5, 7, 25913} one can take C(p) = 24. Bibliography:

J. Garnek. On p-degree of elliptic curves International Journal of Number Theory, 2018. J. Garnek. On class numbers of division ﬁelds of abelian varieties. J. Th´eor. Nombres Bordeaux, 31(1):227–242, 2019. J. Garnek. Equivariant splitting of the Hodge–de Rham exact sequence. arXiv e-prints, page arXiv:1904.05074, April 2019. Thank you for your attention!

Local torsion (n, d)-degree

Definition Let A/K be an abelian variety. The (n, d)-degree of A is:

Dn,d (A/K) := min{[L : K]: A(L) contains a subgroup d isomorphic to (Z/n) }.

Conjecture (David & Weston; Gamzon) Let A be an abelian variety over Q with an endomorphism ring that embeds into a totally real ﬁeld. Then:

lim Dp,1(A/ p) = ∞, p→∞ Q

where the limit is taken over prime numbers. Definition (universal deformation ring) univ RE,p – ring that parameterizes all lifts of ρE,p to artinian local rings with residue ﬁeld Fp.

Question univ ∼ How often RE,p = Zp[[x, y, z]]?

Mazur: if for every [K : Qp] = 2 one has E(K)[p] = 0, then univ ∼ RE,p = Zp[[x, y, z]]. Connection to the local torsion conjecture

class numbers of division ﬁelds of abelian varieties:

Hiranouchi gave a lower bound for #Cl(Kn) for an elliptic curve with surjective ρE,p and E(Qp)[p] = 0.

equivariant splitting of the Hodge–de Rham exact sequence:

I tried to find curves with no ”canonical lift”, i.e. for which the canonical lift is not a jacobian. This could have helped to find answer to the conjecture on canonical lifts. Let: s0 = 0, s1 = 1, sk+2 = 4sk+1 − sk . 2 2 100 000 Then (sk+1 + sk )k=1 is a prime iff

k ∈ {1, 2, 3, 4, 5, 131, 200, 296, 350, 519, 704, 950, 5598,

6683, 7445, 8775, 8786, 11565, 12483}.

we still don’t know whether there are infinitely many primes in the Fibbonacci sequence! One expects 2n − 1 to be prime infinitely often, but 2n + 1 to be prime only finitely often, (Borel-Cantelli lemma + Prime Number Theorem) 2n + 9262111 is never a prime. See: https://mathoverflow.net/q/258322 Equivariant splitting... Conjecture Let char k = p > 0 and let X be a smooth normal curve with an action of a finite p-group G. Denote Y := X /G. Then:

0 ∼ 0 M 0 H (ΩX /k ) = k[G] · H (ΩY /k ) ⊕ H (ΩXP /k ) P∈X M ⊕ ker( IGP,0 → k[G]) P∈X 1 ∼ 1 M 1 HdR (X /k) = k[G] · HdR (Y /k) ⊕ HdR (XP /k) P∈X M ⊕2 ⊕ ker( IGP,0 → k[G]) P∈X

1 (as k[G]-modules), where XP → P are some GP,0-covers, ”locally isomorphic” to X → X /GP,0. Moreover, H1 (X /k) =∼ I ⊕N for some N. dR P GP,2

n Proven for ”singular covers” and for G = Z/p . Let: X : y p − y = xp+1, Then:

p p+1 p2−1 ∼ 3 Aut X = {σbc : b − b = c , c · (c + 1) = 0} = E(p )

(Heisenberg group – cardinality p3, exponent p), where:

1/p σbc (x, y) = (x + c, y + b + c x).

1 ∼ ⊕N One checks that HdR (X /k) = IG . Why the method of proof fails for higher cohomology?

i • G i • G We want to compare H (Y , (F ) ) and H (Y , F ) using two spectral sequences:

ij i j • i+j G • I E2 = H (Y , H (G, F )) ⇒ R Γ (F ) ij i j • i+j G • II E2 = H (G, H (Y , F )) ⇒ R Γ (F ).

For i = 1: Why the method of proof fails for higher cohomology?

i • G i • G We want to compare H (Y , (F ) ) and H (Y , F ) using two spectral sequences:

ij i j • i+j G • I E2 = H (Y , H (G, F )) ⇒ R Γ (F ) ij i j • i+j G • II E2 = H (G, H (Y , F )) ⇒ R Γ (F ).

For i > 1: Proposition If X is ordinary and G acts on X , then this action is weakly ramiﬁed.

Proof WLOG G = Z/p. Let λX := p-rank of X , π : X → X /G =: Y , R := ramiﬁcation divisor of π. By Deuring–Shafarevich formula:

λX − 1 = p · (λY − 1) + #R· (p − 1)

By Riemann–Roch:

2(gX − 1) = p · 2(gY − 1) + deg R. Proposition If X is ordinary and G acts on X , then this action is weakly ramiﬁed.

Proof If X is ordinary, then:

gX − 1 = λX − 1 = p · (λY − 1) + #Supp(R) · (p − 1)

¬ p · (gY − 1) + #R· (p − 1) 1 = (g − 1) − deg R + #R· (p − 1). X 2 Thus: X X X (#GP,i − 1) ¬ 2(p − 1) P i P which implies GP,2 = 0 for all P ∈ Supp(R). Converse theorems

Theorem (J.G.) If the action of G is weakly ramiﬁed then the sequence

0 G 1 G 1 G 0 → H (X , ΩX /k ) → HdR (X /k) → H (X , OX ) → 0

is exact also on the right. Proof: method of proof of Main Theorem. Converse theorems

Theorem The Hodge–de Rham sequence splits provided that any of the following conditions holds: (1) the action of G on X is weakly ramiﬁed and the p-Sylow subgroup of G is cyclic,

(2) the action of G on X lifts to W2(k), (3) X is ordinary. Proof:

1 Previous theorem and group-theoretical considerations.

2 An equivariant version of Deligne-Illusie theorem.

3 The Hodge–de Rham sequences for X and Jac X are the same.

We can lift the action of G on Jac X to W2(k) using Serre-Tate canonical lift. Applications

Corollary

If the action of G on X lifts to W2(k), it must be weakly ramiﬁed.

Corollary Any group action on an ordinary curve must be weakly ramiﬁed. Idea of the proof of MT

Main ideas show that the „defect”

0 G 1 G δ := dimk H (X , ΩX /k ) + dimk H (X , OX ) 1 G − dimk HdR (X /k)

is positive, 1 • G 1 • G compare H (F ) and H ((F ) ) for

• • • F = π∗ΩX /k and F = π∗OX ,

where π : X → Y := X /G is the quotient morphism. Group cohomology of sheaves

Let π : X → Y := X /G be the quotient morphism. Consider G-sheaves on Y (sheaves with an action of G). Definition Sheaf of G-invariants of F:

F G (U) := F(U)G .

Group cohomology of a G-sheaf:

Hi (G, F) := Ri (−)G (F). Properties of group cohomology

Properties: 1 if Y = Spec(A), F = Mf then:

Hi (G, F) = Hi^(G, M).

i i 2 H (G, F)Q = H (G, FQ ),

3 if OX (D) is G-invariant then for i > 0 the sheaf

i H (G, π∗OX (D))

on Y is supported on the wild ramiﬁcation locus of π. Two spectral sequences

Grothendieck’s spectral sequence applied to the commutative diagram:

(−)G G-sheaves on Y sheaves on Y Γ(Y ,−) Γ(Y ,−) (−)G k[G] -mod k -mod .

yields:

ij i j • i+j G • I E2 = H (Y , H (G, F )) ⇒ R Γ (F ) ij i j • i+j G • II E2 = H (G, H (Y , F )) ⇒ R Γ (F ).

We want to compare

10 1 • G I E2 = H (Y , (F ) ) 01 1 • G II E2 = H (Y , F ) . Local–global formula

Low degree exact sequences and properties of group cohomology of sheaves ⇒

δ = dimk Γ(Y , G) 1 1 (where G = im H (G, π∗OX ) → H (G, π∗ΩX /k ) – this is a sheaf with ﬁnite support) X = dimk GQ , Q∈Y

where 1 1 GQ = im H (G, (π∗OX )Q ) → H (G, (π∗ΩX /k )Q )

and the above map is induced by the de Rham diﬀerential. Artin-Schreier covers

We can assume that G = hσi = Z/p.

By Artin-Schreier theory, if P is a ramiﬁcation point of π, Q := π(P), then X looks locally like:

t−np − t−n = x−n

where: t, x are uniformizers at P and Q, σ(t) = t = t − 1 tn+1 + .... (1+tn)1/n n In particular, GP,n = G, GP,n+1 = 0. Computing group cohomology

By taking the long exact sequence for:

0 → k[[t]] → k((t)) → k((t))/k[[t]] → 0

and using Hilbert’s 90th theorem, we see that:

1 1 H (G, (π∗OX )Q ) = H (G, k[[t]]) = coker k((t))G → (k((t))/k[[t]])G .

Proposition 1 The basis of H (G, (π∗OX )Q ) is given by:

i t , for i = −n,..., −1, p - i. Various definitions of Hi (G, F)

Let π : X → Y be a Galois covering of curves, G := Aut(X /Y ). Grothendieck deﬁned two variants of group cohomology for a G-invariant coherent OX -module F: Hi (G, F) := Ri ΓG (X , F), i i G H (G, F) := R π∗ (F). They are connected via a spectral sequence. ”My deﬁnition” of group cohomology for a G-invariant coherent OX -module G: i i G HJG (G, G) := R (−) (G). Proposition For a G-invariant F ∈ OX -mod:

i i HJG (G, π∗F) = π∗(H (G, F)),

(Proof: π∗ is an exact functor on the coherent sheaves) Class numbers Estimate on the class numbers

Let p be a ﬁxed prime number and let K be a number ﬁeld. Let A/K be an abelian variety of dimension d. Denote by r the maximal

possible number of EndK (A)-independent points of A(K). Let also: ( dimFp ker (A[p] → Ap[p]) , if A has good reduction at p, hp := 2d, otherwise. (1) Theorem

∞ kn # Cl(Kn)[p ] = p , where:   X kn 2rd − hp · min {[Kp : Qp] · d, r} · n − C, p|p

where the constant C depends on K, A and p. X X C := 2dr min{αp, βp} + r hpβp + mp. (2) p-p p|p where: ?? Class numbers of division fields of CM AVs

If A/K is an abelian variety with complex multiplication and p is a fixed prime completely split in the CM field, then the tower of fields n Kn := K(A[p ]) forms a Zp-extension. Therefore by Iwasawa theory: µpn+λn+O(1) # Cl(Kn) = p . for some µ, λ ∈ N. µ conjecturally vanishes, Fukuda, K. Komatsu, and S. Yamagata: λ r − d.

Kummer theory of abelian varieties

Suppose that P1,..., Pr ∈ A(K) are EndK (A)-independent. Definition The Kummer extension of Kn: 1 1 L := K P ,..., P . n n pn 1 pn r

(n) n ⊕r Γ : Gal(Ln/Kn) ,→ A[p ] 1 1 σ 7→ σ P − P ,... . pn 1 pn 1 Properties of the Kummer extension

Ln/Kn is abelian,

Ln/Kn may be ramiﬁed only over primes over p and over primes of bad reduction of A, Γ(n) is ”almost an isomorphism”: Theorem (Bashmakov, Ribet)

# coker Γ(n) ¬ C = C(K, A, p) Criterion for surjectivity of Γ(n)

Theorem (J.G.) (n) If Sp2d (Zp) ⊂ ρA,p(GK ), then Γ is an isomorphism. Notation: S K∞ = n Kn, S L∞ = n Ln, (∞) ⊕r Γ : Gal(L∞/K∞) → Tp(A) .

Sketch of the proof: Let N := L1 ∩ K∞. Γ(1) is an isomorphism by Ribet’s work,

study of Gal(N/K1) as Fp[Gal(K1/K)]-module implies:

N = K1. Nakayama’s lemma applied to the diagram: (∞) Γ ⊕r Gal(L∞/K∞) Tp(A)

(1) Γ ⊕r Gal(L1/L1 ∩ K∞) = Gal(L1/K1) A[p] =∼ implies that Γ(∞) is surjective. Sketch of the proof of the Main Theorem

n ⊕r 2d·n·r Kummer theory ⇒ [Ln : Kn] ≈ #A[p ] = p

for p ∈ Spec(OK ):

(n) Ip := hunion of all inertia subgroups in Ln/Kn over pi (n) [ (n) I := h Ip i p

I (n) Ln /Kn is unramiﬁed and abelian ⇒ I (n) Ln is contained in the Hilbert class ﬁeld of Kn ⇒

I (n) (n) 2d·n·r (n) # Cl(Kn) [Ln : Kn] = [Ln : Kn]/#I ≈ p /#I . Estimating the inertia subgroups

Lemma (A) If p - p: ( (n) 1, A has good reduction at p #Ip = O(1), otherwise.

Idea of proof: study of the N´eron model and its group of components over Fp. Lemma (B) If p|p: (n) hp·min{d·[Kp:Qp],r}·n #Ip ¬ p .

n Idea of proof: A(Kp)/p A(Kp) and classiﬁcation theorem of compact p-adic Lie groups. Proof of lemma A

Lemma (A) (n) If p - p then #Ip ¬ C.

Proof: L – maximal unramiﬁed extension of Kp, P ∈ A(L). α m · p – number of geometric components of Ap ⇒

α 0 m · p · Pe ∈ A (Fp) A0 is p-divisible (connected algebraic group) ⇒

m · pα · Pe = pn · Re, kernel of reduction is m-divisible ⇒ pα · P = pn · R0 for some R0 ∈ A(L), Thus:

(n) (n) α ⊕r (n) (n) (n) 2dαr Γ (Ip ) ⊂ A[p ] ⇒ #Ip = #Γ (Ip ) ¬ p . Extended bibliography Extended bibliographyI

S. Arias-de Reyna, W. Gajda, and S. Petersen. Big monodromy theorem for abelian varieties over ﬁnitely generated ﬁelds. J. Pure Appl. Algebra, 217(2):218–229, 2013.

P. Achinger, J. Witaszek, and M. Zdanowicz. Liftability of the Frobenius morphism and images of toric varieties, 2017.

M. I. Bashmakov. The cohomology of abelian varieties over a number ﬁeld. Russian Mathematical Surveys, 27(6):25, 1972.

G. Banaszak, W. Gajda, and P. Krasoń. Detecting linear dependence by reduction maps. J. Number Theory, 115(2):322–342, 2005.

C. Blake. A Deuring criterion for abelian varieties. Bull. Lond. Math. Soc., 46(6):1256–1263, 2014. Extended bibliographyII

S. Bosch, W. Lutkebohmert,¨ and M. Raynaud. N´eron models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990.

J. Bertin and A. Mézard. Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques. Invent. Math., 141(1):195–238, 2000.

P. Cartier. Une nouvelle opération sur les formes différentielles. C. R. Acad. Sci. Paris, 244:426–428, 1957.

Ch.-L. Chai, B. Conrad, and F. Oort. Complex multiplication and lifting problems, volume 195 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2014.

J. B. Cosgrave and K. Dilcher. The Gauss-Wilson theorem for quarter-intervals. Acta Math. Hungar., 142(1):199–230, 2014. Extended bibliographyIII

P. L. Clark and A. Lacy. There are genus one curves of every index over every infinite, finitely generated field. J. Reine Angew. Math., 749:65–86, 2019.

B. Conrad. A modern proof of Chevalley’s theorem on algebraic groups. J. Ramanujan Math. Soc., 17(1):1–18, 2002.

P. L. Clark and X. Xarles. Local bounds for torsion points on abelian varieties. Canad. J. Math., 60(3):532–555, 2008.

D. S. Dummit and R. M. Foote. Abstract algebra. John Wiley & Sons, Inc., Hoboken, NJ, third edition, 2004.

P. Deligne and L. Illusie. Rel`evements modulo p2 et d´ecomposition du complexe de de Rham. Invent. Math., 89(2):247–270, 1987.

L. V. Dieulefait. Explicit determination of the images of the Galois representations attached to abelian surfaces with End(A) = Z. Experiment. Math., 11(4):503–512 (2003), 2002. Extended bibliographyIV

J. de Jong and R. Noot. Jacobians with complex multiplication. In Arithmetic algebraic geometry (Texel, 1989), volume 89 of Progr. Math., pages 177–192. Birkh¨auser Boston, Boston, MA, 1991.

Ch. David and T. Weston. Local torsion on elliptic curves and the deformation theory of Galois representations. Math. Res. Lett., 15(3):599–611, 2008.

T. Fukuda, K. Komatsu, and Sh. Yamagata. Iwasawa λ-invariants and Mordell-Weil ranks of abelian varieties with complex multiplication. Acta Arith., 127(4):305–307, 2007.

G. Faltings, G. Wustholz,¨ F. Grunewald, N. Schappacher, and U. Stuhler. Rational points. Aspects of Mathematics, E6. Friedr. Vieweg & Sohn, Braunschweig, second edition, 1986. Papers from the seminar held at the Max-Planck-Institut fur¨ Mathematik, Bonn/Wuppertal, 1983/1984. Extended bibliographyV

A. Gamzon. Local torsion on abelian surfaces with real multiplication √ by Q( 5). Int. J. Number Theory, 10(7):1807–1827, 2014.

J. Garnek. Arytmetyka krzywych eliptycznych. Master Thesis, 2016.

J. Garnek. On p-degree of elliptic curves. Int. J. Number Theory, 14(3):693–704, 2018.

J. Garnek. Equivariant splitting of the Hodge–de Rham exact sequence. arXiv e-prints, page arXiv:1904.05074, April 2019.

J. Garnek. On class numbers of division ﬁelds of abelian varieties. J. Th´eor. Nombres Bordeaux, 31(1):227–242, 2019.

A. Grothendieck and J. A. Dieudonné. Eléments de géométrie algébrique. III, volume 166 of Pub. Math. IHES. Springer-Verlag, Berlin, 1971. Extended bibliographyVI

B. H. Gross and J. Harris. Real algebraic curves. Annales scientiﬁques de l’Ecole´ Normale Sup´erieure, 14(2):157–182, 1981.

R. Greenberg. Iwasawa theory—past and present. In Class ﬁeld theory—its centenary and prospect (Tokyo, 1998), volume 30 of Adv. Stud. Pure Math., pages 335–385. Math. Soc. Japan, Tokyo, 2001.

A. Grothendieck. Sur quelques points d’alg`ebre homologique. Tˆohoku Math. J. (2), 9:119–221, 1957.

Ch. Hall. An open-image theorem for a general class of abelian varieties. Bull. Lond. Math. Soc., 43(4):703–711, 2011. With an appendix by Emmanuel Kowalski.

R. Hartshorne. On the De Rham cohomology of algebraic varieties. Inst. Hautes Etudes´ Sci. Publ. Math., (45):5–99, 1975.

R. Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. Extended bibliographyVII

T. Hiranouchi. Local torsion primes and the class numbers associated to an elliptic curve over Q. Hiroshima Math. J., 49(1):117–127, 2019. R. Hortsch. On the canonical representation of curves in positive characteristic. New York J. Math., 18:911–924, 2012.

M. Hindry and J. H. Silverman. Diophantine geometry, volume 201 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. An introduction. N. Ishii. Trace of Frobenius endomorphism of an elliptic curve with complex multiplication. Bulletin of the Australian Mathematical Society, 70(1):125–142, 2004.

N. Jacobson. Basic algebra. I. W. H. Freeman and Company, New York, second edition, 1985.

Q. Ji and H. Qin. CM elliptic curves and primes captured by quadratic polynomials. Asian J. Math., 18(4):707–726, 2014. Extended bibliographyVIII

N. M. Katz. Serre-Tate local moduli. In Algebraic surfaces (Orsay, 1976–78), volume 868 of Lecture Notes in Math., pages 138–202. Springer, Berlin-New York, 1981.

B. K¨ock. Galois structure of Zariski cohomology for weakly ramiﬁed covers of curves. Amer. J. Math., 126(5):1085–1107, 2004.

A. Kontogeorgis. On the tangent space of the deformation functor of curves with automorphisms. Algebra Number Theory, 1(2):119–161, 2007. M. Kronberg, M. A. Soomro, and J. Top. Twists of elliptic curves. SIGMA Symmetry Integrability Geom. Methods Appl., 13:Paper No. 083, 13, 2017.

B. K¨ock and J. Tait. On the de-Rham cohomology of hyperelliptic curves. Res. Number Theory, 4(2):4:19, 2018.

E. E. Kummer. Uber¨ die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. Reine Angew. Math., 44:93–146, 1852. Extended bibliographyIX

K.-W Lan. Arithmetic compactiﬁcations of PEL-type Shimura varieties, volume 36 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2013.

M. Larsen. Maximality of Galois actions for compatible systems. Duke Math. J., 80(3):601–630, 1995.

The LMFDB Collaboration. The l-functions and modular forms database. http://www.lmfdb.org, 2020. [Online; accessed 13 March 2020].

B. Mazur. An ”infinite fern” in the universal deformation space of Galois representations. Collect. Math., 48(1-2):155–193, 1997. Journées Arithmétiques (Barcelona, 1995).

W. Messing. The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Mathematics, Vol. 264. Springer-Verlag, Berlin-New York, 1972. Extended bibliographyX

J. S. Milne. Etale´ cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.

J. S. Milne. Abelian varieties (v2.00), 2008. Available at www.jmilne.org/math/.

G. Malle and D. Testerman. Linear algebraic groups and ﬁnite groups of Lie type, volume 133 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2011.

D. Mumford. Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics. Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition.

S. Nakajima. Action of an automorphism of order p on cohomology groups of an algebraic curve. J. Pure Appl. Algebra, 42(1):85–94, 1986. Extended bibliographyXI

J. Neukirch. Algebraic number theory, volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.

T. Oda. The ﬁrst de Rham cohomology group and Dieudonn´e modules. Ann. Sci. Ecole´ Norm. Sup. (4), 2:63–135, 1969.

A. Ogus. Hodge Cycles and Crystalline Cohomology, pages 357–414. Springer Berlin Heidelberg, Berlin, Heidelberg, 1981.

T. Ohshita. Asymptotic lower bound of class numbers along a Galois representation. Journal of Number Theory, 211:95 – 112, 2020.

F. Oort. CM-liftings of abelian varieties. J. Algebraic Geom., 1(1):131–146, 1992. Extended bibliographyXII

R. Pries. A short guide to p-torsion of abelian varieties in characteristic p. In Computational arithmetic geometry, volume 463 of Contemp. Math., pages 121–129. Amer. Math. Soc., Providence, RI, 2008.

R. Pries and H. J. Zhu. The p-rank stratiﬁcation of Artin-Schreier curves. Ann. Inst. Fourier (Grenoble), 62(2):707–726, 2012.

2 3 H. Qin. Anomalous primes of the elliptic curve ED : y = x + D. Proc. Lond. Math. Soc. (3), 112(2):415–453, 2016.

M. Raynaud. Sch´emas en groupes de type (p,..., p). Bull. Soc. Math. France, 102:241–280, 1974.

K. A. Ribet. Kummer theory on extensions of abelian varieties by tori. Duke Math. J., 46(4):745–761, 1979.

J. J. Rotman. An introduction to homological algebra. Universitext. Springer, New York, second edition, 2009. Extended bibliographyXIII

J.-P. Serre. Local ﬁelds, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg.

J.-P. Serre. Abelian l-adic representations and elliptic curves. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, second edition, 1989. With the collaboration of Willem Kuyk and John Labute.

J.-P. Serre. Lie algebras and Lie groups, volume 1500 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, second edition, 1992. 1964 lectures given at Harvard University.

J.-P. Serre. Oeuvres/Collected papers. IV. 1985–1998. Springer Collected Works in Mathematics. Springer, Heidelberg, 2013. Reprint of the 2000 edition. Extended bibliography XIV

S. S. Shatz. Group schemes, formal groups, and p-divisible groups. In Arithmetic geometry (Storrs, Conn., 1984), pages 29–78. Springer, New York, 1986.

J. H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.

J. H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer, Dordrecht, second edition, 2009. Ju. V. Sosnovski˘ı. Commutator structure of symplectic groups. Mat. Zametki, 24(5):641–648, 734, 1978.

The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2016.

H. Stichtenoth. Algebraic function ﬁelds and codes. Universitext. Springer-Verlag, Berlin, 1993. Extended bibliographyXV

D. Subrao. The p-rank of Artin-Schreier curves. Manuscripta Math., 16(2):169–193, 1975.

A. V. Sutherland. Computing images of Galois representations attached to elliptic curves. Forum Math. Sigma, 4:e4, 79, 2016.

F. Sairaiji and T. Yamauchi. On the class numbers of the ﬁelds of the pn-torsion points of certain elliptic curves over Q. J. Number Theory, 156:277–289, 2015.

F. Sairaiji and T. Yamauchi. On the class numbers of the ﬁelds of the pn-torsion points of elliptic curves over Q. J. Th´eor. Nombres Bordeaux, 30(3):893–915, 2018.

I. D. Supunenko and A. E. Zalesskii. Reduced symmetric powers of natural realizations of the groups SLm(p) and Spm(p) and their restrictions to subgroups. Siberian Math. J., 31(4):33–46. Extended bibliography XVI

J. Tate. Modular Forms and Fermat’s Last Theorem, chapter Finite Flat Group Schemes, pages 121–154. Springer New York, New York, NY, 1997.

Ch. Towse. Weierstrass points on cyclic covers of the projective line. Trans. Amer. Math. Soc., 348(8):3355–3378, 1996.

C. Voisin. Hodge theory and complex algebraic geometry. I, volume 76 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps.

T. Wedhorn. De Rham cohomology of varieties over fields of positive characteristic. In Higher-dimensional geometry over finite fields, volume 16 of NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., pages 269–314. IOS, Amsterdam, 2008. Extended bibliography XVII

Ch. A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.

Yu. G. Zarhin. A ﬁniteness theorem for unpolarized abelian varieties over number ﬁelds with prescribed places of bad reduction. Invent. Math., 79(2):309–321, 1985.