= Solution
An <isogeny of elliptic curves> $\phi:E_1\to E_2$ over a <field> $K$ is a nonconstant morphism preserving the identity and addition. It is <surjective> and finite: its image is a closed connected subgroup, and a nonconstant morphism of <smooth projective curves> is finite. The <degree of an isogeny> is the degree of its induced extension of <function fields>. A <separable isogeny> has degree equal to the number of geometric kernel points; in general the degree is the length of the finite kernel scheme, incorporating inseparability. The <degree of an isogeny> multiplies under composition, and degree one means an <isomorphism>. In characteristic zero all <elliptic isogenies> are separable. In positive characteristic the <Frobenius isogeny> illustrates inseparability, while an <elliptic isogeny> of degree <prime> to the characteristic is separable.
The <homomorphism group of elliptic curves> includes the zero homomorphism, the constant identity map, and every nonzero member is an <elliptic isogeny>. The operations are pointwise addition and negation; for endomorphisms, composition also gives the <endomorphism ring of an elliptic curve>. We put $q(0)=0$ and $q(\phi)=\deg\phi$ otherwise. We now prove the <positive definiteness of degree on elliptic-curve homomorphisms> in full.
Let $E=E_2$, and choose a <Weierstrass model>. Its coordinate $x$ has a double pole at $O$, and the generic fibre of $x$ consists of $P$ and $-P$. On $E\times E$ the rational function $x(P)-x(Q)$ therefore has <Weil divisor>
$$
\Delta+\Delta_- -2V-2H,
$$
where $\Delta=\{P=Q\}$, $\Delta_-=\{P=-Q\}$, $V=\{O\}\times E$, and $H=E\times\{O\}$. The generic zeros on the two diagonals have multiplicity one, and the generic poles on $V,H$ have multiplicity two; their intersections have codimension two and add no <Weil divisor> components. This argument also uses the general Weierstrass involution in characteristic $2$, where $-P$ need not mean changing just the sign of $y$.
For the sum and difference maps $a(P,Q)=P+Q$ and $s(P,Q)=P-Q$, the divisors $a^*(O)$ and $s^*(O)$ are $\Delta_-$ and $\Delta$. Consequently, for the degree-one <line bundle> $L=\mathcal O_E(O)$,
$$
a^*L\otimes s^*L\cong\operatorname{pr}_1^*(L^{\otimes2})\otimes\operatorname{pr}_2^*(L^{\otimes2}).
$$
Pull back along $P\mapsto(\phi(P),\psi(P))$ and take degrees of <line bundles> on $E_1$. A nonconstant map pulls back $L$ to degree equal to its degree, while a constant map pulls it back to a trivial bundle of degree zero. We obtain the <degree parallelogram law>
$$
\boxed{q(\phi+\psi)+q(\phi-\psi)=2q(\phi)+2q(\psi).}
$$
Using the line-bundle identity is important: it still restricts correctly when $\phi=\psi$ or $\phi=-\psi$, when restriction of the displayed rational function would be identically zero.
Since $q(-\phi)=q(\phi)$, applying the identity to $n\phi$ and $\phi$ gives the second-difference recursion
$$
q((n+1)\phi)+q((n-1)\phi)=2q(n\phi)+2q(\phi).
$$
Starting from $q(0)=0$ and $q(\phi)$ yields $q(n\phi)=n^2q(\phi)$ for all <integers> $n$. Define
$$
B(\phi,\psi)=\frac{q(\phi+\psi)-q(\phi-\psi)}4
=\frac{q(\phi+\psi)-q(\phi)-q(\psi)}2.
$$
It is symmetric and odd in each variable. Applying the parallelogram identity to the four terms in its first expression gives
$$
B(u+w,v)+B(u-w,v)=2B(u,v).
$$
Interchanging $u,w$ and using oddness also gives $B(u+w,v)-B(u-w,v)=2B(w,v)$. Adding proves $B(u+w,v)=B(u,v)+B(w,v)$; symmetry proves additivity in the other variable. Thus $q$ is a <quadratic form> with associated rational <bilinear form> $B$.
Every nonzero homomorphism has positive degree, so $q(\phi)>0$ for $\phi\ne0$. The scaling identity also shows that the homomorphism <group> is a <torsion-free group>. Hence $q$ extends to its rational <vector space>. On the span of any finitely many rationally independent homomorphisms, the matrix of $B$ is rational. Positivity on nonzero rational vectors implies nonnegativity on real vectors by density. If the real matrix had a nontrivial kernel, its rational linear equations would have a nonzero rational solution, contradicting positivity. Thus its real extension is a <positive-definite quadratic form>. This establishes the requested positivity as well as the quadratic identities.
An important companion to degree is the <dual isogeny>. Under $E\cong\operatorname{Pic}^0(E)$ via $P\mapsto[P-O]$, the pullback $\phi^*:\operatorname{Pic}^0(E_2)\to\operatorname{Pic}^0(E_1)$ defines $\widehat\phi:E_2\to E_1$. Divisor pushforward corresponds to $\phi$, and pushforward after pullback multiplies <divisor classes> by $n=\deg\phi$. Therefore $\phi\widehat\phi=[n]$. Composing shows that $\widehat\phi\phi-[n]$ takes values in the finite kernel of $\phi$; a morphism from a connected smooth curve into this finite scheme must be constant, and its identity value is zero. Thus
$$
\boxed{\widehat\phi\phi=[n],\qquad\phi\widehat\phi=[n],\qquad\deg\widehat\phi=n.}
$$
The last equality follows from $\deg[n]=n^2$, already proved by applying $q(m\phi)=m^2q(\phi)$ to the identity map. The <dual isogeny> is unique, since the difference of two choices composed with $\phi$ would vanish and $\phi$ is <surjective>. It follows that being isogenous is an equivalence relation: duality supplies symmetry, and composition supplies transitivity. For a finite subgroup of geometric points of order <prime> to the characteristic, the quotient curve and its quotient map give an <elliptic isogeny> with that kernel; in general one uses finite subgroup schemes. This is the geometric source of explicit descent maps, including the <two-isogeny formula> in Question 4.
Finally <elliptic isogenies> preserve the rational <rank of an abelian group> over a <number field>. Their dual compositions are multiplication by $n$, so after tensoring the <rational point> <groups> with $\mathbb Q$ the <elliptic isogeny> becomes invertible. For a single curve, its endomorphism algebra in characteristic zero is $\mathbb Q$ or an imaginary quadratic <field>, giving ordinary <integer> multiplication or <complex multiplication>. In positive characteristic a supersingular curve can instead have a quaternionic endomorphism algebra. The positive degree form controls the sizes of these homomorphisms, while duality makes their arithmetic and geometric properties accessible.
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