HasseWeil.lean`

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7 declarations, 3 contain sorry, 0 declared as axioms

The Hasse--Weil conjecture for elliptic curves

The $L$-series $L(E, s)$ of an elliptic curve $E$ over a number field converges absolutely on $\operatorname{Re} s > 3/2$ ([Tate1966], p. 416; [Gross2011], Lecture 2, §1). The Hasse--Weil conjecture predicts that it has an analytic continuation to the whole complex plane and satisfies a functional equation ([Tate1966], p. 416; [Gross2011], Conjecture 2.7, stated there for the completed $L$-function). Only the continuation is stated here. Over $\mathbb{Q}$ it follows from the modularity theorem.

[Wikipedia] states the conjecture for the Hasse--Weil zeta function and asks only for a meromorphic continuation, which is equivalent to a meromorphic continuation of $L(E, s)$. This is also the hypothesis under which the Birch and Swinnerton-Dyer conjecture is stated ([Gross2011], Conjecture 2.10), so the weaker continuation is recorded here as well.

References:

Wikipedia
[Tate1966] John Tate. "On the conjectures of Birch and Swinnerton-Dyer and a geometric analog." Seminaire Bourbaki, Vol. 9, Exp. No. 306 (1966), 415-440, numdam
[Gross2011] Benedict H. Gross. "Lectures on the conjecture of Birch and Swinnerton-Dyer." Arithmetic of L-functions, IAS/Park City Math. Ser. 18, AMS (2011), 169-209, math.harvard.edu

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Millennium/BSD.lean

def HasseWeil.HasMeromorphicContinuation Documented declaration: HasMeromorphicContinuation line 56 source
Source documentation
The $L$-series of E has the meromorphic continuation L: L is meromorphic on $\mathbb{C}$ and agrees with E.LSeries on $\operatorname{Re} s > 3/2$, where the series converges absolutely.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, WeierstrassCurve, Complex
Proof uses: And, Meromorphic, LT.lt, Real, HDiv.hDiv, OfNat.ofNat, Complex.re, Eq, WeierstrassCurve.LSeries
15 instances, coercions and auxiliary constants left out.
```
def HasMeromorphicContinuation (E : WeierstrassCurve K) (L : ℂ → ℂ) : Prop :=
…
```
def HasseWeil.HasAnalyticContinuation Documented declaration: HasAnalyticContinuation line 62 source
Source documentation
The $L$-series of E has the analytic continuation L: L is analytic on $\mathbb{C}$ and agrees with E.LSeries on $\operatorname{Re} s > 3/2$, where the series converges absolutely.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, WeierstrassCurve, Complex
Proof uses: And, AnalyticAt, LT.lt, Real, HDiv.hDiv, OfNat.ofNat, Complex.re, Eq, WeierstrassCurve.LSeries
15 instances, coercions and auxiliary constants left out.
```
def HasAnalyticContinuation (E : WeierstrassCurve K) (L : ℂ → ℂ) : Prop :=
…
```
theorem HasseWeil.HasAnalyticContinuation.hasMeromorphicContinuation An analytic continuation is a meromorphic continuation line 67 source
Source documentation
An analytic continuation is a meromorphic continuation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, WeierstrassCurve, Complex, HasseWeil.HasAnalyticContinuation, HasseWeil.HasMeromorphicContinuation
Proof uses: And.intro, Meromorphic, LT.lt, Real, HDiv.hDiv, OfNat.ofNat, Complex.re, Eq, WeierstrassCurve.LSeries, AnalyticAt.meromorphicAt, And.left, AnalyticAt, And.right
15 instances, coercions and auxiliary constants left out.
```
theorem HasAnalyticContinuation.hasMeromorphicContinuation {L : ℂ → ℂ}
    (hL : HasAnalyticContinuation E L) : HasMeromorphicContinuation E L :=
…
```
Mentions: HasseWeil.HasAnalyticContinuation, HasseWeil.HasMeromorphicContinuation
theorem HasseWeil.HasMeromorphicContinuation.unique A meromorphic continuation is unique away from isolated points: two meromorphic continuations of the $L$-series of E agree on a punctured neighbourhood of every point line 77 source
Source documentation
A meromorphic continuation is unique away from isolated points: two meromorphic continuations of the $L$-series of E agree on a punctured neighbourhood of every point. They may differ at isolated points, since Meromorphic does not constrain the value of a function at any single point.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, WeierstrassCurve, Complex, HasseWeil.HasMeromorphicContinuation, Filter.EventuallyEq, nhdsWithin, Compl.compl, Set, Singleton.singleton
Proof uses: Eq, WithTop, Int, meromorphicOrderAt, HSub.hSub, OfNat.ofNat, Nat, Top.top, Iff.mpr, Filter.Eventually, meromorphicOrderAt_eq_top_iff, Filter.eventually_of_mem, Set.ofPred, LT.lt, Real, HDiv.hDiv, Complex.re, nhdsWithin_le_nhds, IsOpen.mem_nhds, Complex.isOpen_re_gt and 37 more
85 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem HasMeromorphicContinuation.unique {L L' : ℂ → ℂ}
    (hL : HasMeromorphicContinuation E L) (hL' : HasMeromorphicContinuation E L') (x : ℂ) :
    L =ᶠ[𝓝[≠] x] L' := by
…
```
Mentions: HasseWeil.HasMeromorphicContinuation
theorem HasseWeil.exists_hasMeromorphicContinuation The $L$-series of an elliptic curve over a number field has a meromorphic continuation to $\mathbb{C}$ sorry line 92 source
Source documentation
The $L$-series of an elliptic curve over a number field has a meromorphic continuation to $\mathbb{C}$. This is the Hasse--Weil conjecture in the form stated in [Wikipedia], a weak form of [Gross2011], Conjecture 2.7, and the hypothesis under which the Birch and Swinnerton-Dyer conjecture is stated ([Gross2011], Conjecture 2.10).

depends on sorry what this means
Statement uses: Field, NumberField, WeierstrassCurve, WeierstrassCurve.IsElliptic, Exists, Complex, HasseWeil.HasMeromorphicContinuation
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_hasMeromorphicContinuation (E : WeierstrassCurve K) [E.IsElliptic] :
    ∃ L, HasMeromorphicContinuation E L := by
…
```
Mentions: HasseWeil.HasMeromorphicContinuation
theorem HasseWeil.exists_hasAnalyticContinuation The Hasse--Weil conjecture ([Tate1966], p sorry line 100 source
Source documentation
The Hasse--Weil conjecture ([Tate1966], p. 416; [Gross2011], Conjecture 2.7): the $L$-series of an elliptic curve over a number field has an analytic continuation to $\mathbb{C}$.

depends on sorry what this means
Statement uses: Field, NumberField, WeierstrassCurve, WeierstrassCurve.IsElliptic, Exists, Complex, HasseWeil.HasAnalyticContinuation
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_hasAnalyticContinuation (E : WeierstrassCurve K) [E.IsElliptic] :
    ∃ L, HasAnalyticContinuation E L := by
…
```
Mentions: HasseWeil.HasAnalyticContinuation
theorem HasseWeil.exists_hasAnalyticContinuation_rat The Hasse--Weil conjecture over $\mathbb{Q}$: the $L$-series of an elliptic curve over $\mathbb{Q}$ has an analytic continuation to $\mathbb{C}$ sorry line 110 source
Source documentation
The Hasse--Weil conjecture over $\mathbb{Q}$: the $L$-series of an elliptic curve over $\mathbb{Q}$ has an analytic continuation to $\mathbb{C}$. This follows from the modularity theorem, see [Wikipedia].

depends on sorry what this means
Statement uses: WeierstrassCurve, Rat, WeierstrassCurve.IsElliptic, Exists, Complex, HasseWeil.HasAnalyticContinuation
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_hasAnalyticContinuation_rat (E : WeierstrassCurve ℚ) [E.IsElliptic] :
    ∃ L, HasAnalyticContinuation E L := by
…
```
Mentions: HasseWeil.HasAnalyticContinuation