Yes, there is a connection between fixed point theory and number theory. Fixed point theory is a part of number theory and is used to study the behavior of certain functions. Number theory is a branch of mathematics that studies the properties of integers and the relationships between them. Fixed point theory is used to study the properties of functions and their fixed points, which are points that stay the same when the function is applied. This can be used to investigate properties of integers and other mathematical objects.

Here are a reference related to the connection between fixed point theory and number theory

"Fixed Points and Periodic Points in Number Theory and Algebraic Geometry" by K. Ebrahimi-Fard, L. Guo, and D. Manchon, Found. Comput. Math. 18, 63 (2018).

This paper provides an overview of the connection between fixed point theory and number theory, with a focus on applications to algebraic geometry.

I would recommend to see the references of the paper for older references.

I hope this helps.

The answer may be somewhat highbrow and I must admit that I do not know the details of most of what I'm about to write but as an algebraic geometer, this comes to mind.

First, in algebraic topology, a fundamental construction is that of the (singular) cohomology of a topological space. In the case of a compact manifold X, singular cohomology (with rational coefficients) produces groups H^i(X,Q) that measure a number of things about X. E.g., when X is a surface (without boundary), these groups recover the genus and thus the homeomorphism type of X (this favourable situation is exceptional, in general, you don't get so much information out of cohomology). Perhaps the most important property of cohomology, certainly one that's essential in computations, is that it is functorial: a continuous map f:X->Y induces morphisms f^*:H^i(Y,Q)->H^i(X,Q) of Q-vector spaces for all i and the association f->f^* behaves well with respect to composition. Since we work with compact manifolds, the Q-vector spaces H^i(X,Q) are actually of finite dimension; hence if f:X->X is a continuous map, f^*:H^i(X,Q)->H^i(X,Q), as a linear endomorphism of a finite-dimensional Q-vector space, possesses a trace.

Lefschetz proved a beautiful theorem relating the (weighted) number of fixed points of such a map f with isolated fixed points with these traces. Namely say that f has isolated fixed points if given a fixed point x of f, there exists an open subset U containing x such that f has no fixed points in U\{x}. In other words, the set of fixed points of f is discrete or, equivalently, finite since X is compact. Now given a fixed point x, since X is a manifold, there is an open neighbourhood U of x that is homeomorphic to a ball in Euclidean space and given an open neighbourhood V of x in f^{-1)(U) with the same property, the map V\{x}->U\{x} may be viewed as a continuous map from B\{0} to B\{0} with B a ball centered at the origin in R^n and thus, up to homotopy, as a continuous map g of the sphere S^{n-1} of dimension n-1 to itself. It then has a well-defined Brouwer degree i(f,x) which is an integer that only depends on f (and x) and is none other than the scale of the homothety g^*:H^{n-1}(S^{n-1},Q)->H^{n-1}(S^{n-1},Q)—the Q-vector space H^{n-1}(S^{n-1},Q) is indeed of dimension one, so that you do get a homothety (it is not clear that this number is an integer; it would be cleaner to work with integral coefficients for which this is obvious but there are too many details missing to care at this point, I think). If you don't know any homotopy theory, it does not matter much: simply remember that one can associate with an isolated fixed point of f an integer i(f,x). Wikipedia states the version below as the Lefschetz–Hopf theorem but as far as I know, at least in algebro-geometric circles, it's known as the Lefschetz trace formula.

Theorem (Lefschetz's trace formula). Let f:X->X be a continuous map from a compact manifold X of dimension d to itself; assume that the set Fix(f) of fixed points is finite. Then the sum of the integers i(f,x) for x ranging through Fix(f) is equal to the sum from i=0 to i=d of (-1)^iTr(f^*:H^i(X,Q)->H^i(X,Q)) where Tr denotes the trace.

This theorem is wonderful because it directly relates the number of fixed points of f to a construction of linear algebra and is especially useful because cohomology lends itself to computation. For example, if the alternated sum of traces is non-zero, then f must have a fixed po|/-int. One can use this to prove Brouwer's theorem on fixed points of endomorphisms of the closed ball: if X is a closed ball, then H^i(X,Q)=0 for i different from 0 hence the alternating sum on the right is 1 and f must have a fixed point.

Now the following striking idea, due to Weil, attempts to import these techniques to algebraic geometry. In this context, you work with an algebraic variety X, defined by homogeneous equations in n+1 variables with coefficients in a field k which we will take finite with q elements. Thus X is the "set of points" in projective space P^n that satisfy the given equations. It is however essential to consider not only the solutions with coordinates in the field k but also the solutions with coordinates in K for any field extension K/k: they form a set which we denote by X(K). For example, X(k) is the set of honest solutions in P^n(k), that is, with coordinates in k. In this context, it actually suffices to consider the solutions with coordinates in a given algebraic closure k^a/k of k.

One way to study X is to consider the following combinatorial problem. There is exactly one subextension K/k of k^a/k which is such that K is a field with q^n elements: set k_n=K. The set X(K) is a subset of P^n(K) which is finite as K is a finite field. One of Weil's insights is that the cardinality |X(K)| of X(K) is a very interesting invariant of X. You might want to study the asymptotics as n tends to infinity or even combine them into a generating function: this is what Weil did to formulate his famous conjectures. He considers the power series Z_X(t)=exp(\sum|X(k)^n|/n.t^n) (where the sum ranges from 1 to infinity) and makes several conjectures, one of which being that this power series is actually a rational function, with explicit numerator and denominator described as products of polynomials whose degree he can control via other topological invariants actually given by cohomology as above (I'll come back to that later), at least in favourable cases. Some assumptions have to be made on X, namely that X is irreducible and (more importantly) smooth, that is, sufficiently nice, but they're too technical to bother with here.

Not content with the conjectures he'd proposed, Weil set out a path to solve them, building on the ideas of Lefschetz. The field k^n can actually be described as a set of fixed points, namely the fixed points of the n-th power of the Frobenius ring automorphism f of k^a that sends x to x^q. The Frobenius also induces an automorphism of projective space P^n, sending a point [x_0:...:x_n]=x to f(x)=[x_0^q:...:x_n^q], and X is stable under the action of this automorphism: this is because the equations defining X are with coefficients in k and are thus fixed by the Frobenius f, so that f(x) is a solution of these equations if x is a solution. We then have an automorphism f:X->X of X, and X(k^n) is precisely the set of fixed points of the n-th power f^n of f.

Weil's idea is now that if we were able to construct a cohomology theory for algebraic varieties, rather than topological spaces, that would have a sufficient number of properties common with singular cohomology, then Lefschetz's trace formula should also be true in that context and should allow us to compute or at least understand |X(k^n)|. This turned out to be true and led to Grothendieck's étale and l-adic cohomology for schemes—algebraic varieties are an example of schemes—as part of a massive undertaking of foundational and investigative work he did in the 50's and 60's.

Grothendieck's theory of schemes is more refined than the classical algebraic geometry of algebraic varieties, however. In particular, it allows one to consider varieties over the integers Z, defined by polynomial equations with coefficients in Z rather than a field like Q or a finite field like in the classical setting. As opposed to a variety over a finite field, a variety over the integers has the distinct advantage of that considering solutions with coordinates in the complex numbers makes sense: we call these solutions complex points. If the variety X is nice (smooth), then the set X(C) of complex points has the structure of a complex analytic manifold which is compact if X is projective, that is, given as above by homogeneous polynomials. Singular cohomology makes sense and is well-behaved for X(C) and thus the Q-vector spaces H^i(X(C),Q) have a dimension b_i(X(C)): this is the i-th Betti number of X—Betti cohomology is another name for singular cohomology. Given a variety X over a field with a prime number of elements, X can be "lifted" to a variety Y over the integers and part of Weil's conjectures is that in sufficiently nice cases, Z_X(t) is a rational function, the numerator and denominator are products of polynomials and the degrees of these polynomials are given by the Betti numbers of Y(C). (I think the basic argument is that there are comparison theorems between l-adic cohomology and singular cohomology but again, at this point, this whole answer is too imprecise for this to matter...)

Thank you Samuel for your time! I have studied algebraic topology and cohomology theory, and can understand most part of your answer. Thanks again for such details!